INTRODUCTION TO ALGEBRAIC TOPOLOGY
doc. RNDr. Martin ˇCadek, CSc.
Faculty of Science of Masaryk University
Department of Mathematics and Statistics
Technical cooperation: Service Center for E-learning,
Faculty of Informatics, Masaryk University, Brno 2013
c 2013 Masaryk University
1
0. Foreword
These notes form a brief overview of basic topics in a usual introductory course
of algebraic topology. They were prepared for my series of lectures at the Okayama
University in 2002 and rewritten in 2013. They cannot substitute standard textbooks.
The technical proofs of several important theorems are omitted and many other theorems
are not proved in full generality. However, in all such cases I have tried to give
references to well known textbooks the list of which you can find at the end.
I would like to express my acknowledgements to my former student Richard Lastovecki
whose comments helped me to correct and improve the text.
The notes are available online in electronic form at
http://is.muni.cz/el/1431/jaro2013/M8130/um/39015882/
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Contents
0. Foreword 1
1. Basic notions and constructions 3
2. CW-complexes 9
3. Simplicial and singular homology 14
4. Homology of CW-complexes and applications 24
5. Singular cohomology 31
6. More homological algebra 37
7. Products in cohomology 45
8. Vector bundles and Thom isomorphism 52
9. Poincar´e duality 57
10. Homotopy groups 67
11. Fundamental group 74
12. Homotopy and CW-complexes 79
13. Homotopy excision and Hurewicz theorem 85
14. Short overview of some further methods in homotopy theory 93
References 99
3
1. Basic notions and constructions
1.1. Notation. The closure, the interior and the boundary of a topological space
X will be denoted by X, int X and ∂X, respectively. The letter I will stand for the
interval [0, 1]. Rn
and Cn
will denote the vector spaces of n-tuples of real and complex
numbers, respectively, with the standard norm x = n
i=1 |xi|2
. The sets
Dn
= {x ∈ Rn
; x ≤ 1},
Sn
= {x ∈ Rn+1
; x = 1}
are the n-dimensional disc and the n-dimensional sphere, respectively.
1.2. Categories of topological spaces. Every category consists of objects and
morphisms between them. Morphisms f : A → B and g : B → C can be composed in
a morphism g ◦ f : A → C and for every object B there is a morphism idB : B → B
such that idB ◦f = f and g ◦ idB = g.
The category with topological spaces as objects and continuous maps as morphisms
will be denoted Top. Topological spaces with distinquished points (usually denoted by
∗) and continuous maps f : (X, ∗) → (Y, ∗) such that f(∗) = ∗ form the category Top∗.
Topological spaces X, A will be called a pair of topological spaces if A is a subspace
of X (notation (X, A)). The notation f : (X, A) → (Y, B) means that f : X → Y is a
continuous map which preserves subspaces, i. e. f(A) ⊆ B. The category Top2
consists
of pairs of topological spaces as objects and continuous maps f : (X, A) → (Y, B) as
morphisms. Finally, Top2
∗ will denote the category of pairs of topological spaces with
distinquished points in subspaces and continuous maps preserving both subspaces and
distinquished points.
The right category for doing algebraic topology is the category of compactly generated
spaces. We will not go into details and refer to Chapter 5 of [May]. In fact, the
majority of spaces we deal with in this text are compactly generated.
From now on, a space will mean a topological space and a map will mean a continuous
map.
1.3. Homotopy. Maps f, g : X → Y are called homotopic, notation f ∼ g, if there
is a map h : X × I → Y such that h(x, 0) = f(x) and h(x, 1) = g(x). This map is
called homotopy between f and g. The relation ∼ is an equivalence. Homotopies in
categories Top∗, Top2
or Top2
∗ have to preserve distinquished points, i. e. h(∗, t) = ∗,
subsets or both subsets and distinquished points, respectively.
Spaces X and Y are called homotopy equivalent if there are maps f : X → Y and
g : Y → X such that f ◦ g ∼ idY and g ◦ f ∼ idX. We also say that the spaces X and
Y have the same homotopy type. The maps f and g are called homotopy equivalences.
A space is called contractible if it is homotopy equivalent to a point.
Example. Sn
and Rn+1
− {0} are homotopy equivalent. As homotopy equivalences
take the inclusion f : Sn
→ Rn+1
− {0} and g : Rn+1
− {0} → Sn
, g(x) = x/ x .
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1.4. Retracts and deformation retracts. Let i : A → X be an inclusion. We say
that A is a retract of X if there is a map r : X → A such that r ◦ i = idA. The map r
is called a retraction.
We say that A is a deformation retract of X (sometimes also strong deformation
retract) if i ◦ r : X → A → X is homotopic to the identity on X relative to A, i.e.
there is a homotopy h : X × I → X such that h(−, 0) = idX, h(−, 1) = i ◦ r and
h(i(−), t) = idA for all t ∈ I. The map h is called a deformation retraction .
Exercise A. Show that deformation retract of X is homotopy equivalent to X.
1.5. Basic constructions in Top. Consider a topological space X with an equivalence
. Then X/ is the set of equivalence classes with the topology determined
by the projection p : X → X/ in the following way: U ⊆ X/ is open iff p−1
(U) is
open in X.
Exercise A. The map f : (X/ ) → Y is continuous iff the composition f ◦ p : X →
(X/ ) → Y is continuous.
We will show this constructions in several special cases. Let A be a subspace of X.
The quotient X/A is the space X/ where x y iff x = y or both x and y are elements
of A. This space is often considered as a based space with base point determined by
A. If A = ∅ we put X/∅ = X ∪ {∗}.
Exercise B. Prove that Dn
/Sn−1
is homeomorphic to Sn
. For it consider f : Dn
→ Sn
f(x1, x2, . . . , xn) = (2 1 − x 2x, 2 x 2
− 1).
Disjoint union of spaces X and Y will be denoted X Y . Open sets are unions of
open sets in X and in Y . Let A be a subspace of X and let f : A → Y be a map.
Then X ∪f Y is the space (X Y )/ where the equivalence is generated by relations
a f(a).
The mapping cylinder of a map f : X → Y is the space
Mf = X × I ∪f×1 Y
which arises from X × I and Y after identification of points (x, 1) ∈ X × I and
f(x) ∈ Y .
Mf
f(X)
f
X
Y
Figure 1.1. Mapping cylinder
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Exercise C. We have two inclusions iX : X = X × {0} → Mf and iY : Y → Mf and
a retraction r : Mf → Y . How is r defined?
X
f
~~}}}}}}}}
iX

f
AAAAAAAA
Y iY
// Mf r
// Y
Prove that
(1) Y is a deformation retract of Mf ,
(2) iX ◦ r = f,
(3) iY ◦ f ∼ iX.
The mapping cone of a mapping f : X → Y is the space
Cf = Mf /(X × {0}).
A special case of a mapping cone is the cone of a space X
CX = X × I/(X × {0}) = CidX
.
The suspension of a space X is the space
SX = CX/(X × {1}).
Exercise D. Show that SSn
= Sn+1
. For it consider the map f : Sn
× I → Sn+1
f(x, t) = ( 1 − (2t − 1)2x, 2t − 1).
The join of spaces X and Y is the space
X Y = X × Y × I/
where is the equivalence generated by (x, y1, 0) (x, y2, 0) and (x1, y, 1) (x2, y, 1).
Exercise E. Show that the join operation is associative and compute the joins of two
points, two intervals, several points, S0
X, Sn
Sm
.
1.6. Basic constructions in Top∗ and Top2
. Let X be a space with a base point
x0. The reduced suspension of X is the space
ΣX = SX/({x0} × I)
with base point determined by x0 × I. In the next section in 2.8 we will show that
ΣX is homotopy equivalent to SX.
The space
(X, x0) ∨ (Y, y0) = X × {y0} ∪ {x0} × Y
with distinquished point (x0, y0) is called the wedge of X and Y and usually denoted
only as X ∨ Y .
The smash product of spaces (X, x0) and (Y, y0) is the space
X ∧ Y = X × Y/(X × {y0} ∪ {x0} × Y ) = X × Y/X ∨ Y.
Analogously, the smash product of pairs (X, A) and (Y, B) is the pair
(X × Y, A × Y ∪ X × B).
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Exercise A. Show that Sm
∧ Sn
= Sn+m
. One way how to do it is to prove that
X/A ∧ Y/B ∼= X × Y/A × Y ∪ X × B.
1.7. Homotopy extension property. We say that a pair of topological spaces
(X, A) has the homotopy extension property (abbreviation HEP) if any map f : X → Y
and any homotopy h : A × I → Y such that h(a, 0) = f(a) for a ∈ A, and
f ∪ h : X × {0} ∪ A × I → Y
is continuous, can be extended to a homotopy H : X×I → Y such that H(x, 0) = f(x)
and H(a, t) = h(a, t) for all x ∈ X, a ∈ A and t ∈ I, i.e. H is an arrow making the
diagram
X × {0} ∪ A × I
f∪h
//

Y
X × I
H
77ppppppp
commutative. If the pair (X, A) satisfies HEP, we call the inclusion A → X a cofibra-
tion.
A × I
X × I
H
h
f
Y
X × {0}
Figure 1.2. Homotopy extension property
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Theorem. A pair (X, A) has HEP if and only if X ×{0}∪A×I is a retract of X ×I.
A × I
X × {0}
Figure 1.3. Retraction X × I → X × {0} ∪ A × I
Exercise A. Using this Theorem show that the pair (Dn
, Sn−1
) satisfies HEP. Many
other examples will be given in the next section.
X
y
Z = r(Z)
r(y)
r(x)
Figure 1.4. Retraction D1
× I → D1
× {0} ∪ S0
× I
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Proof of Theorem. Let (X, A) has HEP. Put Y = X × {0} ∪ A × I and consider
f ∪ h : X × {0} ∪ A × I → X × {0} ∪ A × I to be an identity. Its extension
H : X × I → X × {0} ∪ A × I is a retraction.
Let r : X × I → X × {0} ∪ A × I be a retraction. Given a map f and a homotopy
h as in the definition which together determine a continuous map F = (f ∪ h) :
X × {0} ∪ A × I → Y , then H = F ◦ r is an extension of f ∪ h.
Exercise B. Let a pair (X, A) satisfy HEP and consider a map g : A → Y . Prove
that (X ∪g Y, Y ) also satisfies HEP.
Exercise C. Let X be a Hausdorff compact space and let an inclusion A → X is a
cofibration. Prove that A is a closed subset of X.
Exercise D. Consider the closed subset set A = {1/n ∈ R; n = 0, 1, 2, . . . } ∪ {0} of
the interval [0, 1]. However, the inclusion A → [0, 1] is not a cofibration. Prove it.
Exercise E. Let Mf be a mapping cylinder of a map f : X → Y . Show that the
inclusion iX : X → Mf is a cofibration. In particular, the map f : X → Y can be
factored into the composition r◦iX of the cofibration iX and the homotopy equivalence
r. (See the exercise after the definition of the mapping cylinder.)
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2. CW-complexes
2.1. Constructive definition of CW-complexes. CW-complexes are all the spaces
which can be obtained by the following construction:
(1) We start with a discrete space X0
. Single points of X0
are called 0-dimensional
cells.
(2) Suppose that we have already constructed Xn−1
. For every element α of an
index set Jn take a map fα : Sn−1
= ∂Dn
α → Xn−1
and put
Xn
=
α
Xn−1
∪fα Dn
α .
Interiors of discs Dn
α are called n-dimensional cells and denoted by en
α.
(3) We can stop our construction for some n and put X = Xn
or we can proceed
with n to infinity and put
X =
∞
n=0
Xn
.
In the latter case X is equipped with inductive topology which means that
A ⊆ X is closed (open) iff A ∩ Xn
is closed (open) in Xn
for every n.
Example A. The sphere Sn
is a CW-complex with one cell e0
in dimension 0, one
cell en
in dimension n and the constant attaching map f : Sn−1
→ e0
.
Example B. The real projective space RPn
is the space of 1-dimensional linear subspaces
in Rn+1
. It is homeomorhic to
Sn
/(v −v) ∼= Dn
/(w −w), for w ∈ ∂Dn
= Sn−1
.
However, Sn−1
/(w −w) ∼= RPn−1
. So RPn
arises from RPn−1
by attaching one ndimensional
cell using the projection f : Sn−1
→ RPn−1
. Hence RPn
is a CW-complex
with one cell in every dimension from 0 to n.
We define RP∞
= ∞
n=1 RPn
. It is again a CW-complex.
Example C. The complex projective space CPn
is the space of complex 1-dimensional
linear subspaces in Cn+1
. It is homeomorhic to
S2n+1
/(v λv) ∼= {(w, 1 − |w|2) ∈ Cn+1
; w ≤ 1}/((w, 0) λ(w, 0), w = 1)
∼= D2n
/(w λw; w ∈ ∂D2n
)
for all λ ∈ C, |λ| = 1. However, ∂D2n
/(w λw) ∼= CPn−1
. So CPn
arises from CPn−1
by attaching one 2n-dimensional cell using the projection f : S2n−1
= ∂D2n
→ CPn−1
.
Hence CPn
is a CW-complex with one cell in every even dimension from 0 to 2n.
Define CP∞
= ∞
n=1 CPn
. It is again a CW-complex.
2.2. Another definition of CW-complexes. Sometimes it is advantageous to be
able to describe CW-complexes by their properties. We carry it out in this paragraph.
Then we show that the both definitions of CW-complexes are equivalent.
Definition. A cell complex is a Hausdorff topological space X such that
10
(1) X as a set is a disjoint union of cells eα
X =
α∈J
eα.
(2) For every cell eα there is a number, called dimension.
Xn
=
dim eα≤n
eα
is the n-skeleton of X.
(3) Cells of dimension 0 are points. For every cell of dimension ≥ 1 there is a
characteristic map
ϕα : (Dn
, Sn−1
) → (X, Xn−1
)
which is a homeomorphism of int Dn
onto eα.
The cell subcomplex Y of a cell complex X is a union Y = α∈K eα , K ⊆ J, which
is a cell complex with the same characterictic maps as the complex X.
A CW-complex is a cell complex satisfying the following conditions:
(C) Closure finite property. The closure of every cell belongs to a finite subcomplex,
i. e. subcomplex consisting only from a finite number of cells.
(W) Weak topology property. F is closed in X if and only if F ∩ ¯eα is closed for
every α.
Example. Examples of cell complexes which are not CW-complexes:
(1) S2
where every point is 0-cell. It does not satisfy property (W).
(2) D3
with cells e3
= int B3
, e0
x = {x} for all x ∈ S2
. It does not satisfy (C).
(3) X = {1/n; n ≥ 1} ∪ {0} ⊂ R. It does not satisfy (W).
(4) X = ∞
n=1{x ∈ R2
; x − (1/n, 0) = 1/n} ⊂ R2
. If it were a CW-complex, the
set {(1/n, 0) ∈ R2
; n ≥ 1} would be closed in X, and consequently in R2
.
2.3. Equivalence of definitions.
Proposition. The definitions 2.1 and 2.2 of CW-complexes are equivalent.
Proof. We will show that a space X constructed according to 2.1 satisfies definition
2.2. The proof in the opposite direction is left as an exercise to the reader.
The cells of dimension 0 are points of X0
. The cells of dimension n are interiors of
discs Dn
α attached to Xn−1
with charakteristic maps
ϕα : (Dn
α, Sn−1
α ) → (Xn−1
∪fα Dn
α, Xn−1
)
induced by identity on Dn
α. So X is a cell complex. From the construction 2.1 it
follows that X satisfies property (W). It remains to prove property (C). We will carry
it out by induction.
Let n = 0. Then e0
α = e0
α.
Let (C) holds for all cells of dimension ≤ n − 1. en
α is a compact set (since it is an
image of Dn
α). Its boundary ∂en
α is compact in Xn−1
. Consider the set of indices
K = {β ∈ J; ∂en
α ∩ eβ = ∅}.
11
If we show that K is finite, from the inductive assumption we get that ¯en
α lies in a
finite subcomplex which is a union of finite subcomplexes for ¯eβ, β ∈ K.
Choosing one point from every intersection ∂en
α ∩ eβ, β ∈ K we form a set A. A is
closed since any intersection with a cell is empty or a onepoint set. Simultaneously, it
is open, since every its element a forms an open subset (for A − {a} is closed). So A
is a discrete subset in the compact set ∂en
α, consequently, it is finite.
2.4. Compact sets in CW complexes.
Lemma. Let X be a CW-complex. Then any compact set A ⊆ X lies in a finite
subcomplex, particularly, there is n such that A ⊆ Xn
.
Proof. Consider the set of indices
K = {β ∈ J; A ∩ eβ = ∅}.
Similarly as in 2.3 we will show that K is a finite set. Then A ⊆ β∈K ¯eβ and every
¯eβ lies in a finite subcomplexes. Hence A itself is a subset of a finite subcomplex.
2.5. Cellular maps. Let X and Y be CW-complexes. A map f : X → Y is called
a cellular map if f(Xn
) ⊆ Y n
for all n. In Section 5 we will prove that every map
g : X → Y is homotopic to a cellular map f : X → Y . If moreover, g restricted to a
subcomplex A ⊂ X is already cellular, f can be chosen in such a way that f = g on
A.
2.6. Spaces homotopy equivalent to CW-complexes. One can show that every
open subset of Rn
is a CW-complex. In [Hatcher], Theorem A.11, it is proved that
every retract of a CW-complex is homotopy equivalent to a CW-complex. These
two facts imply that every compact manifold with or without boundary is homotopy
equivalent to a CW-complex. (See [Hatcher], Corollary A.12.)
2.7. CW complexes and HEP. The most important result of this section is the
following theorem:
Theorem. Let A be a subcomplex of a CW-complex X. Then the pair (X, A) has the
homotopy extension property.
Proof. According to the last theorem in Section 1 it is sufficient to prove that X ×
{0} ∪ A × I is a retract of X × I. We will prove that it is even a deformation retract.
There is a retraction rn : Dn
× I → Dn
× {0} ∪ Sn−1
× I. (See Section 1.) Then
hn : Dn
× I × I → Dn
× I defined by
hn(x, s, t) = (1 − t)(x, s) + trn(x, s)
is a deformation retraction, i.e. a homotopy between id and rn.
Put Y −1
= A, Y n
= Xn
∪ A. Using hn we can define a deformation retraction
Hn : Y n
× I × I → Y n
× I for the retract Y n
× {0} ∪ Y n−1
× I of Y n
× I. Now define
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the deformation retraction H : X × I × I → X × I for the retract X × {0} ∪ A × I
succesively on the subspaces X ×{0}×I ∪Y n
×I ×I with values in X ×{0}∪Y n
×I.
For n = 0 put
H(x, s, t) = (x, s) for (x, s) ∈ X × {0} or t ∈ [0, 1/2],
H(x, s, t) = H0(x, s, 2(t − 1/2)) for x ∈ Y 0
and t ∈ [1/2, 1].
Suppose that we have already defined H on X ×{0}∪Y n−1
×I. On X ×{0}∪Y n
×I
we put
H(x, s, t) = (x, s) for (x, s) ∈ X × {0} or t ∈ [0, 1/2n+1
],
H(x, s, t) = Hn(x, s, 2n+1
(t − 1/2n+1
)) for x ∈ Y n
and t ∈ [1/2n+1
, 1/2n
],
H(x, s, t) = H(H(x, s, 1/2n
), t) for x ∈ Y n
and t ∈ [1/2n
, 1].
H : X×I×I → X×I is continuous since so are its restrictions on X×{0}×I∪Y n
×I×I
and the space X × I × I is a direct limit of the subspaces X × {0} × I ∪ Y n
× I × I.
X × I
t = 1
8
t = 1
4
t = 1
2
t = 1t = 0
X × {0} ∪ A × I
X × {0} ∪ Y 2
× I
X × {0} ∪ Y 1
× I
X × {0} ∪ Y 0
× I
Figure 2.1. Image of H depending on t
2.8. First criterion for homotopy equivalence.
Proposition. Suppose that a pair (X, A) has the homotopy extension property and
that A is contractible (in A). Then the canonical projection q : X → X/A is a
homotopy equivalence.
Proof. Since A is contractible, there is a homotopy h : A × I → A between idA and
constant map. This homotopy together with idX : X → X can be extended to a
homotopy f : X × I → X. Since f(A, t) ⊆ A for all t ∈ I, there is a homotopy
˜f : X/A × I → X/A such that the diagram
X × I
f
//
q

X
q

X/A × I ˜f
// X/A
13
commutes. Define g : X/A → X by g([x]) = f(x, 1). Then idX ∼ g ◦ q via the
homotopy f and idX/A ∼ q ◦ g via the homotopy ˜f. Hence X is homotopy equivalent
to X/A.
Exercise A. Using the previous criterion show that S2
/S0
∼ S2
∨ S1
.
Exercise B. Using the previous criterion show that the suspension and the reduced
suspension of a CW-complex are homotopy equivalent.
2.9. Second criterion for homotopy equivalence.
Proposition. Let (X, A) be a pair of CW-complexes and let Y be a space. Suppose
that f, g : A → Y are homotopic maps. Then X ∪f Y and X ∪g Y are homotopy
equivalent.
Proof. Let F : A×I → Y be a homotopy between f and g. We will show that X ∪f Y
and X ∪g Y are both deformation retracts of (X × I) ∪F Y . Consequently, they have
to be homotopy equivalent.
We construct a deformation retraction in two steps.
(1) (X × {0}) ∪f Y is a deformation retract of (X × {0} ∪ A × I) ∪F Y .
(2) (X × {0} ∪ A × I) ∪F Y is a deformation retract of (X × I) ∪F Y .
Exercise. Let (X, A) be a pair of CW-complexes. Suppose that A is a contractible
in X, i. e. there is a homotopy F : A → X between idX and const. Using the first
criterion show that X/A ∼= X ∪ CA/CA ∼ X ∪ CA. Using the second criterion prove
that X ∪ CA ∼ X ∨ SA. Then
X/A ∼ X ∨ SA.
Apply it to compute Sn
/Si
, i < n.
14
3. Simplicial and singular homology
3.1. Exact sequences. A sequence of homomorphisms of Abelian groups or modules
over a ring
. . .
fn+1
−−−→ An
fn
−−→ An−1
fn−1
−−−→ An−2
fn−2
−−−→ . . .
is called an exact sequence if
Im fn = Ker fn−1.
Exactness of the following sequences
O −→ A
f
−−→ B, B
g
−−→ C −→ 0, 0 −→ C
h
−−→ D −→ 0
means that f is a monomorphism, g is an epimorphism and h is an isomorphism,
respectively.
A short exact sequence is an exact sequence
0 −→ A
i
−→ B
j
−→ C −→ 0.
In this case C ∼= B/A. We say that the short exact sequence splits if one of the
following three equivalent conditions is satisfied:
(1) There is a homomorphism p : B → A such that pi = idA.
(2) There is a homomorphism q : C → B such that jq = idC.
(3) There are homomorphisms p : B → A and q : C → B such that ip + qj = idB.
The last condition means that B ∼= A ⊕ C with isomorphism (p, q) : B → A ⊕ C.
Exercise. Prove the equivalence of (1), (2) and (3).
3.2. Chain complexes. The chain complex (C, ∂) is a sequence of Abelian groups
(or modules over a ring) and their homomorphisms indexed by integers
. . .
∂n+2
−−−→ Cn+1
∂n+1
−−−→ Cn
∂n
−−→ Cn−1
∂n−1
−−−→ . . .
such that
∂n−1∂n = 0.
This conditions means that Im ∂n ⊆ Ker ∂n−1. The homomorphism ∂n is called a
boundary operator. A chain homomorphism of chain complexes (C, ∂C
) and (D, ∂D
) is
a sequence of homomorphisms of Abelian groups (or modules over a ring) fn : Cn → Dn
which commute with the boundary operators
∂D
n fn = fn−1∂C
n .
3.3. Homology of chain complexes. The n-th homology group of the chain complex
(C, ∂) is the group
Hn(C) =
Ker ∂n
Im ∂n+1
.
The elements of Ker ∂n = Zn are called cycles of dimension n and the elements of
Im ∂n+1 = Bn are called boundaries (of dimension n). If a chain complex is exact,
then its homology groups are trivial.
15
The component fn of the chain homomorphism f : (C, ∂C
) → (D, ∂D
) maps cycles
into cycles and boundaries into boundaries. It enables us to define
Hn(f) : Hn(C) → Hn(D)
by the prescription Hn(f)[c] = [fn(c)] where [c] ∈ Hn(C∗) and [fn(c)] ∈ Hn(D∗
) are
classes represented by the elements c ∈ Zn(C) and fn(c) ∈ Zn(D), respectively.
3.4. Long exact sequence in homology. A sequence of chain homomorphisms
. . . −→ A
f
−−→ B
g
−−→ C −→ . . .
is exact if for every n ∈ Z
. . . −→ An
fn
−−→ Bn
gn
−−→ Cn −→ . . .
is an exact sequence of Abelian groups.
Theorem. Let 0 → A
i
−→ B
j
−→ C → 0 be a short exact sequence of chain complexes.
Then there is a connecting homomorphism ∂∗ : Hn(C) → Hn−1(A) such that the
sequence
. . .
∂∗
−−→ Hn(A)
Hn(i)
−−−→ Hn(B)
Hn(j)
−−−−→ Hn(C)
∂∗
−−→ Hn−1(A)
Hn−1(i)
−−−−−→ . . .
is exact.
Proof. Define the connecting homomorphism ∂∗. Let [c] ∈ Hn(C) where c ∈ Cn is a
cycle. Since j : Bn → Cn is an epimorphism, there is b ∈ Bn such that j(b) = c.
Further, j(∂b) = ∂j(b) = ∂c = 0. From exactness there is a ∈ An−1 such that
i(a) = ∂b. Since i(∂a) = ∂i(a) = ∂∂b = 0 and i is a monomorphism, ∂a = 0 and a is
a cycle in An−1. Put
∂∗[c] = [a].
Now we have to show that the definition is correct, i. e. independent of the choice of c
and b, and to prove exactness. For this purpose it is advantageous to use an appropriate
diagram. It is not difficult and we leave it as an exercise to the reader.
3.5. Chain homotopy. Let f, g : C → D be two chain homomorphisms. We say
that they are chain homotopic if there are homomorphisms sn : Cn → Dn+1 such that
∂D
n+1sn + sn−1∂C
n = fn − gn for all n.
The relation to be chain homotopic is an equivalence. The sequence of maps sn is
called a chain homotopy.
Theorem. If two chain homomorphism f, g : C → D are chain homotopic, then
Hn(f) = Hn(g).
Exercise. Prove the previous theorem from the definitions.
16
3.6. Five Lemma. Consider the diagram
A //
f1 ∼=

B //
f2 ∼=

C //
f3

D //
f4 ∼=

E
f5 ∼=

¯A // ¯B // ¯C // ¯D // ¯E
If the horizontal sequences are exact and f1, f2, f4 and f5 are isomorphisms, then f3
is also an isomorphism.
Exercise. Prove 5-lemma.
3.7. Simplicial homology. We describe two basic ways how to define homology
groups for topological spaces – simplicial homology which is closer to geometric intuition
and singular homology which is more general. For the definition of simplicial
homology we need the notion of ∆-complex, which is a special case of CW-complex.
Let v0, v1, . . . , vn be points in Rm
such that v1 − v0, v2 − v0, vn − v0 are linearly independent.
The n-simplex [v0, v1, . . . , vn] with the vertices v0, v1, . . . , vn is the subspace
of Rm
{
n
i=0
tivi;
n
i=1
ti = 1, ti ≥ 0}
with a given ordering of vertices. A face of this simplex is any simplex determined by
a proper subset of vertices in the given ordering.
Let ∆α, α ∈ J be a collection of simplices. Subdivide all their faces of dimension
i into sets Fi
β. A ∆-complex is a quotient space of disjoint union α∈J ∆α obtained
by identifying simplices from every Fi
β into one single simplex via affine maps which
preserve the ordering of vertices. Thus every ∆-complex is determined only by combinatorial
data.
A special case of ∆-complex is a finite simplicial complex. It is a union of simplices
the vertices of which lie in a given finite set of points {v0, v1, . . . , vn} in Rm
such that
v1 − v0, v2 − v0, . . . , vn − v0 are linearly independent.
Example. Torus, real projective space of dimension 2 and Klein bottle are ∆-complexes
as one can see from the following pictures.
b
b
b
b
b
b
a a a a a a
c c c
Figure 3.1. Torus, RP2
and Klein bottle as ∆-complexes
17
In all the cases we have two sets F2
whose elements are triangles, three sets F1
every with two segments and one set F0
containing all six vertices of both triangles.
These surfaces are also homeomorhic to finite simplicial complexes, but their structure
as simplicial complexes is more complicated than their structure as ∆-complexes.
To every ∆-complex X we can assign the chain complex (C, ∂) where Cn(X) is a free
Abelian group generated by n-simplices of X (i. e. the rank of Cn(X) is the number
of the sets Fn
and the boundary operator on generators is given by
∂[v0, v1, . . . , vn] =
n
i=0
(−1)i
[v0, . . . , ˆvi . . . , vn].
Here the symbol ˆvi means that the vertex vi is omitted. Prove that ∂∂ = 0.
The simplicial homology groups of ∆-complex X are the homology groups of the
chain complex defined above. Later, we will show that these groups are independent
of ∆-complex structure.
Exercise. Compute simplicial homology of S2
(find a ∆-complex structure), RP2
,
torus and Klein bottle (with ∆-complex structures given in example above).
Let X and Y be two ∆-complexes and f : X → Y a map which maps every simplex
of X into a simplex of Y and it is affine on all simplexes. Using appropriate sign
conventions we can define the chain homomorphism fn : Cn(X) → Cn(Y ) induced by
the map f. This chain map enables us to define homomorphism of simplicial homology
groups induced by f.
Having a ∆-subcomplex A of a ∆-complex X (i. e. subspace of X formed by some of
the simplices of X) we can define simplicial homology groups Hn(X, A). The definition
is the same as for singular homology in paragraph 3.9. These groups fit into the long
exact sequence
· · · → Hn(A) → Hn(X) → Hn(X, A) → Hn−1(A) → . . .
See again 3.9.
3.8. Singular homology. The standard n-simplex is the n-simplex
∆n
= {(t0, t1, . . . , tn) ∈ Rn+1
;
n
i=0
ti = 1; ti ≥ 0}.
The j-th face of this standard simplex is the (n−1)-dimensional simplex [e0, . . . , ˆej, . . . , en]
where ej is the vertex with all coordinates 0 with the exception of the j-th one which
is 1. Define
εj
n : ∆n−1
→ ∆n
as the affine map εj
n(t0, t1, . . . , tn−1) = (t0, . . . , tj−1, 0, tj, . . . , tn−1) which maps
e0 → e0, . . . , ej−1 → ej−1, ej → ej+1, . . . , en−1 → en.
It is not difficult to prove
Lemma. εk
n+1εj
n = εj+1
n+1εk
n for k < j.
18
A singular n-simplex in a space X is a continuous map σ : ∆n
→ X. Denote the
free Abelian group generated by all the singular n-simplices by Cn(X) and define the
boundary operator ∂n : Cn(X) → Cn−1(X) by
∂n(σ) =
n
i=0
(−1)i
σεi
n
for n ≥ 0. Put Cn(X) = 0 for n < 0. Using the lemma above one can show that
∂n+1∂n = 0.
The chain complex (Cn, ∂n) is called the singular chain complex of the space X. The
singular homology groups Hn(X) of the space X are the homology groups of the chain
complex (Cn(X), ∂n), i. e.
Hn(X) =
Ker ∂n
Im ∂n+1
.
Next consider a map f : X → Y . Define the chain homomorhism Cn(f) : Cn(X) →
Cn(Y ) on singular n-simplices as the composition
Cn(f)(σ) = fσ.
From definitions it is easy to show that these homomorphisms commute with boundary
operators. Hence this chain homomorphism induces homomorphisms
f∗ = Hn(f) : Hn(X) → Hn(Y ).
Moreover, Hn(idX) = idHn(X) and Hn(fg) = Hn(f)Hn(g). It means that Hn is a
functor from the category Top to the category Ab of Abelian groups and their homomorphisms.
This functor is the composition of the functor C from Top to chain
complexes and the n-th homology functor from chain complexes to abelian groups.
Prove the lemma above and ∂n+1∂n = 0.
Show directly from the definition that the singular homology groups of a point are
H0(∗) = Z and Hn(∗) = 0 for n = 0.
3.9. Singular homology groups of a pair. Consider a pair of topological spaces
(X, A). Then the Cn(A) is a subgroup of Cn(X). Hence we get this short exact
sequence
0 → Cn(A)
i
−→ Cn(X)
j
−→
Cn(X)
Cn(A)
→ 0.
Since the boundary operators in Cn(A) are restrictions of boundary operators in
Cn(X), we can define boundary operators
∂n :
Cn(X)
Cn(A)
→
Cn−1(X)
Cn−1(A)
.
We will denote this chain complex as (C(X, A), ∂) and its homology groups as Hn(X, A).
Notice that the factor Cn(X)/Cn(A) is a free Abelian group generated by singular simplices
σ : ∆n
→ X such that σ(∆n
) A. We will need it later.
19
A map f : (X, A) → (Y, B) induces the chain homomorphism Cn(f) : Cn(X) →
Cn(Y ) which restricts to a chain homomorphism Cn(A) → Cn(B) since f(A) ⊆ B.
Hence we can define the chain homomorphism
Cn(f) : Cn(X, A) → Cn(Y, B)
which in homology induces the homomorphism
f∗ = Hn(f) : Hn(X, A) → Hn(Y, B).
We can again conclude that Hn is a functor from the category Top2
into the category
Ab of Abelian groups. This functor extends the functor defined on the category Top
since every object X and every morphism f : X → Y in Top can be considered as the
object (X, ∅) and the morphism ˆf = f : (X, ∅) → (Y, ∅) in the category Top2
and
Hn(X, ∅) = Hn(X), Hn( ˆf) = Hn(f).
3.10. Long exact sequence for singular homology. Consider inclusions of spaces
i : A → X, i : B → Y and maps j : (X, ∅) → (X, A), j : (Y, ∅) → (Y, B) induced
by idX and idY , respectively. Let f : (X, A) → (Y, B) be a map. Then there are
connecting homomorphisms ∂X
∗ and ∂Y
∗ such that the following diagram
∂X
∗
// Hn(A)
i∗
//
(f/A)∗

Hn(X1
)
j∗
//
f∗

Hn(X, A)
∂X
∗
//
f∗

Hn−1(A)
i∗
//
(f/A)∗
∂Y
∗
// Hn(B)
i∗
// Hn(Y )
j∗
// Hn(Y, B)
∂Y
∗
// Hn−1(B)
i∗
//
commutes and its horizontal sequences are exact.
An analogous theorem holds also for simplicial homology.
Remark. Consider the functor I : Top2
→ Top2
which assigns to every pair (X, A)
the pair (A, ∅). The commutativity of the last square in the diagram above means
that ∂∗ is a natural transformation of functors Hn and Hn−1 ◦ I defined on Top2
.
Proof. We have the following commutative diagram of chain complexes
0 // C(A)
C(i)
//
C(f/A)

C(X)
C(j)
//
C(f)

C(X, A) //
C(f)

0
0 // C(B)
C(i )
// C(Y )
C(j )
// C(Y, B) // 0
with exact horizontal rows. Then Theorem 3.4 and the construction of connecting
homomorphism ∂∗ imply the required statement.
Remark. It is useful to realize how ∂∗ : Hn(X, A) → Hn−1(A) is defined. Every
element of Hn(X, A) is represented by a chain x ∈ Cn(X) with a boundary ∂x ∈
Cn−1(A). This is a cycle in Cn(A) and from the definition in 3.4 we have
∂∗[x] = [∂x].
20
3.11. Homotopy invariance. If two maps f, g : (X, A) → (Y, B) are homotopic,
then they induce the same homomorphisms
f∗ = g∗ : Hn(X, A) → Hn(Y, B).
Proof. We need to prove that the homotopy between f and g induces a chain homotopy
between C∗(f) and C∗(g). For the proof see [Hatcher], Theorem 2.10 and Proposition
2.19 or [Spanier], Chapter 4, Section 4.
Corollary. If X and Y are homotopy equivalent spaces, then
Hn(X) ∼= Hn(Y ).
3.12. Excision Theorem. There are two equivalent versions of this theorem.
Theorem (Excision Theorem, 1st version). Consider spaces C ⊆ A ⊆ X and suppose
that ¯C ⊆ int A. Then the inclusion
i : (X − C, A − C) → (X, A)
induces the isomorphism
i∗ : Hn(X − C, A − C)
∼=
−→ Hn(X, A).
Theorem (Excision Theorem, 2nd version). Consider two subspaces A and B of a
space X. Suppose that X = int A ∪ int B. Then the inclusion
i : (B, A ∩ B) → (X, A)
induces the isomorphism
i∗ : Hn(B, A ∩ B)
∼=
−→ Hn(X, A).
The second version of Excision Theorem holds also for simplicial homology if we
suppose that A and B are ∆-subcomplexes of a ∆-complex X and X = A∪B. In this
case the proof is easy since the inclusion
Cn(i) : Cn(B, A ∩ B) → Cn(A ∪ B, A)
is an isomorphism, namely the both chain complexes are generated by the same n-
simplices.
Exercise. Show that the theorems above are equivalent.
The proof of Excision Theorem for singular homology can be found in [Hatcher],
pages 119 – 124, or in [Spanier], Chapter 4, Sections 4 and 6. The main step (a little
bit technical for beginners) is to prove the following lemma which we will need later.
Lemma. Let U = {Uα; α ∈ J} be a collection of subsets of X such that X =
α∈J int Uα. Denote the free chain complex generated by singular simplices σ with
σ(∆n
) ∈ Uα for some α as CU
n (X). Then
CU
n (X)) → Cn(X)
induces isomorphism in homology.
21
Proof of Excision Theorem. Consider U = {A, B}. Then the inclusion
Cn(i) : Cn(B, A ∩ B) →
CU
n (X)
Cn(A)
is an isomorphism and, moreover, according to the previous lemma, the homology of
the second chain complex is Hn(X, A).
3.13. Homology of disjoint union. Let X = α∈J Xα be a disjoint union. Then
Hn(X) =
α∈J
Hn(Xα).
The proof follows from the definition and connectivity of σ(∆n
) in X for every
singular n-simplex σ.
3.14. Reduced homology groups. For every space X = ∅ we define the augmented
chain complex ( ˜C(X), ˜∂) as follows
˜Cn(X) =
Cn(X) for n = −1,
Z for n = −1.
with ˜∂n = ∂n for n = 0 and ∂0( j
i=1 niσi) = j
i=1 ni. The reduced homology groups
˜Hn(X) are the homology groups of the augmented chain complex. From the definition
it is clear that
˜Hn(X) = Hn(X) for n = 0
and
˜Hn(∗) = 0 for all n.
For pairs of spaces we define ˜Hn(X, A) = Hn(X, A) for all n. Then theorems on long
exact sequence, homotopy invariance and excision hold for reduced homology groups
as well.
Considering a space X with distinguished point ∗ and applying the long exact
sequence for the pair (X, ∗), we get that for all n
˜Hn(X) = ˜Hn(X, ∗) = Hn(X, ∗).
Using this equality and the long exact sequence for unreduced homology we get that
H0(X) ∼= H0(X, ∗) ⊕ H0(∗) ∼= ˜H0(X) ⊕ Z.
Lemma. Let (X, A) be a pair of CW-complexes, X = ∅. Then
˜Hn(X/A) = Hn(X, A)
and we have the long exact sequence
· · · → ˜Hn(A) → ˜Hn(X) → ˜Hn(X/A) → ˜Hn−1(A) → . . .
22
Proof. According to example in Section 2
(X, A) → (X ∪ CA, CA) → (X ∪ CA/CA, ∗) = (X/A, ∗)
is the composition of an excision and a homotopy equivalence. Hence ˜Hn(X/A) =
Hn(X, A). The rest folows from the long exact sequence of the pair (X, A).
Exercise. Prove that ˜Hn( Xα) ∼= ⊕ ˜Hn(Xα).
˜Hn can be considered as a functor from Top∗ to Abelian groups.
3.15. The long exact sequence of a triple. Three spaces (X, B, A) with the
property A ⊆ B ⊆ X are called a triple. Denote i : (B, A) → (X, A) and j : (X, A) →
(X, B) maps induced by the inclusion B → X and idX, respectively. Analogously as
for pairs one can derive the following long exact sequence:
. . .
∂∗
−→ Hn(B, A)
i∗
−→ Hn(X, A)
j∗
−→ Hn(X, B)
∂∗
−→ Hn−1(B, A)
i∗
−→ . . .
3.16. Singular homology groups of spheres. Consider the long exact sequence
of the triple (∆n
, ∂∆n
, Λn−1
= ∂∆n
− ∆n−1
):
· · · → Hi(∆n
, Λn−1
) → Hi(∆n
, ∂∆n
)
∂∗
−→ Hi−1(∂∆n
, Λn−1
) → Hi−1(∆n
, Λn−1
) → . . .
The pair (∆n
, Λn−1
) is homotopy equivalent to (∗, ∗) and hence its homology groups
are zeroes. Next using Excision Theorem and homotopy invariance we get that
Hi(∆n
, Λn−1
) ∼= Hi(∆n−1
, ∂∆n−1
). Consequently, we get an isomorphism
Hi(∆n
, ∂∆n
) ∼= Hi−1(∆n−1
, ∂∆n−1
).
Using induction and computing Hi(∆1
, ∂∆1
) = Hi([0, 1], {0, 1}) ∼= Hi−1({0, 1}, {0})
we get that
Hi(∆n
, ∂∆n
) =
Z for i = n,
0 for i = n.
Doing the induction carefully we can find that the generator of the group Hn(∆n
, ∂∆n
) =
Z is determined by the singular n-simplex id∆n .
The pair (Dn
, Sn−1
) is homeomorphic to (∆n
, ∂∆n
). Hence it has the same homology
groups. Using the long exact sequence for this pair we obtain
˜Hi−1(Sn−1) = Hi(Dn
, Sn−1
) =
0 for i = n,
Z for i = n.
3.17. Mayer-Vietoris exact sequence. Denote inclusions A∩B → A, A∩B → B,
A → X, B → X by iA, iB, jA, jB, respectively. Let C → A ∩ B and suppose that
X = int A ∪ int B. Then the following sequence
. . .
∂∗
−−→ Hn(A ∩ B, C)
(iA∗,iB∗)
−−−−−→ Hn(A, C) ⊕ Hn(B, C)
jA∗−jB∗
−−−−−→ Hn(X, C)
∂∗
−−→ Hn−1(A ∩ B, C) −→ . . .
23
is exact.
Proof. The covering U = {A, B} satisfies conditions of Lemma 3.12. The sequence of
chain complexes
0 −→
C(A ∩ B)
C(C)
i
−→
C(A)
C(C)
⊕
C(B)
C(c)
j
−→
CU
(X)
C(C)
−→ 0
where i(x) = (x, x) and j(x, y) = x − y is exact. Consequently, it induces a long
exact sequence. Using Lemma 3.12 we get that Hn(CU
(X), C(C)) = Hn(X, C), which
completes the proof.
3.18. Equality of simplicial and singular homology. Let (X, A) be a pair of
∆-complexes. Then the natural inclusion of simplicial and singular chain complexes
C∆
(X, A) → C(X, A) induces the isomorphism of simplicial and singular homology
groups
H∆
n (X, A) ∼= Hn(X, A).
Outline of the proof. Consider the long exact sequences for the pair (Xk
, Xk−1
) of
skeletons of X. We get
H∆
n+1(Xk
, Xk−1
) //

H∆
n (Xk−1
) //

H∆
n (Xk
) //

H∆
n (Xk
, Xk−1
) //

H∆
n−1(Xk−1
)

Hn+1(Xk
, Xk−1
) // Hn(Xk−1
) // Hn(Xk
) // Hn(Xk
, Xk−1
) // Hn−1(Xk−1
)
Using induction by k we have H∆
i (Xk−1
) = Hi(Xk−1
) for all i. Further, C∆
i (Xk
, Xk−1
)
is according to definition zero if i = k and free Abelian of rank equal the number of isimplices
∆i
α if i = k. The homology groups H∆
i (Xk
, Xk−1
) have the same description.
Since
α
∆k
α/
α
∂∆k
α = Xk
/Xk−1
we get the isomorphism
H∆
i (Xk
/Xk−1
) → Hi(
α
∆k
α/
α
∂∆k
α) = Hi(Xk
/Xk−1
).
Applying 5-lemma (see 3.6) in the diagram above, we get that H∆
n (Xk
) → Hn(Xk
) is
an isomorphism.
If X is finite ∆-complex, we are ready. If it is not, we have to prove that H∆
n (X) =
Hn(X). See [Hatcher], page 130.
24
4. Homology of CW-complexes and applications
4.1. First applications of homology. Using homology groups we can easily prove
the following statements:
(1) Sn
is not a retract of Dn+1
.
(2) Every map f : Dn
→ Dn
has a fixed point, i.e. there is x ∈ Dn
such that
f(x) = x.
(3) If ∅ = U ⊆ Rn
and ∅ = V ⊆ Rm
are open homeomorphic sets, then n = m.
Outline of the proof. (1) Suppose that there is a retraction r : Dn+1
→ Sn
. Then we
get the commutative diagram
Z = Hn(Sn
)
id //
i∗ ((QQQQQQQQQQQQ
Hn(Sn
) = Z
Hn(Dn+1
) = 0
r∗
66mmmmmmmmmmmm
which is a contradiction.
(2) Suppose that f : Dn
→ Dn
has no fixed point. Then we can define the map
g : Dn
→ Sn−1
where g(x) is the intersection of the ray from f(x) to x with Sn−1
.
However, this map would be a retraction, a contradiction with (1).
(3) The proof of the last statement follows from the isomorphisms:
Hi(U, U−{x}) ∼= Hi(Rn
, Rn
−{x}) ∼= ˜Hi−1(Rn
−{x}) ∼= ˜Hi−1(Sn−1
) =
Z for i = n,
0 for i = n.
4.2. Degree of a map. Consider a map f : Sn
→ Sn
. In homology f∗ : ˜Hn(Sn
) →
Hn(Sn
) has the form
f∗(x) = ax, a ∈ Z.
The integer a is called the degree of f and denoted by deg f.
The degree has the following properties:
(1) deg id = 1.
(2) If f ∼ g, then deg f = deg g.
(3) If f is not surjective, then deg f = 0.
(4) deg(fg) = deg f · deg g.
(5) Let f : Sn
→ Sn
, f(x0, x1, . . . , xn) = (−x0, x1, . . . , xn). Then deg f = −1.
(6) The antipodal map f : Sn
→ Sn
, f(x) = −x has deg f = (−1)n+1
.
(7) If f : Sn
→ Sn
has no fixed point, then deg f = (−1)n+1
.
Proof. We outline only the proof of (5) and (7). The rest is not difficult and left as an
exercise.
We show (5) by induction on n. The generator of ˜H0(S0
) is 1 − (−1) and f∗ maps
it in (−1) − 1. Hence the degree is −1. Suppose that the statement is true for n. To
25
prove it for n+1 we use the diagram with rows coming from a suitable Mayer-Vietoris
exact sequence
0 // ˜Hn+1(Sn+1
)
∼= //
f∗

˜Hn(Sn
) //
(f/Sn)∗

0
0 // ˜Hn+1(Sn+1
)
∼= // ˜Hn(Sn
) // 0
If (f/Sn
)∗ is a multiplication by −1, so is f∗.
To prove (7) we show that f is homotopic to the antipodal map through the homo-
topy
H(x, t) =
tf(x) − (1 − t)x
tf(x) − (1 − t)x
.
Corollary. Sn
has a nonzero continuous vector field if and only if n is odd.
Proof. Let Sn
has such a field v(x). We can suppose v(x) = 1. Then the identity is
homotopic to antipodal map through the homotopy
H(x, t) = cos tπ · x + sin tπ · v(x).
Hence according to properties (2) and (6)
(−1)n+1
= deg(− id) = deg(id) = 1.
Consequently, n is odd.
On the contrary, if n = 2k+1, we can define the required vector field by prescription
v(x0, x1, x2, x3, . . . , x2k, x2k+1) = (−x1, x0, −x3, x2, . . . , −x2k+1, x2k).
Exercise. Prove the properties (3), (4) and (6) of the degree.
4.3. Local degree. Consider a map f : Sn
→ Sn
and y ∈ Sn
such that f−1
(y) =
{x1, x2, . . . , xm}. Let Ui be open disjoint neighbourhoods of points xi and V a neighbourhood
of y such that f(Ui) ⊆ V . Then
(f/Ui)∗ : Hn(Ui, Ui − {xi}) ∼= Hn(Sn
, Sn
− {xi}) = Z
−→ Hn(V, V − {y}) ∼= Hn(Sn
, Sn
− {y}) = Z
is a multiplication by an integer which is called a local degree and denoted by deg f|xi.
Theorem A. Let f : Sn
→ Sn
, y ∈ Sn
and f−1
(y) = {x1, x2, . . . , xm}. Then
deg f =
m
i=1
deg f|xi.
For the proof see [Hatcher], Proposition 2.30, page 136.
The suspension Sf of a map f : X → Y is given by the prescription Sf(x, t) =
(f(x), t).
26
Theorem B. deg Sf = deg f for any map f : Sn
→ Sn
.
Proof. f induces Cf : CSn
→ CSn
. The long exact sequence for the pair (CSn
, Sn
)
and the fact that SSn
= CSn
/Sn
give rise to the diagram
˜Hn+1(Sn+1
) ∼=
//
Sf∗

˜Hn+1(CSn
, Sn
)
∂∗
∼=
//
Cf∗

˜Hn(Sn
)
f∗

˜Hn+1(Sn+1
) ∼=
// ˜Hn+1(CSn
, Sn
)
∂∗
∼=
// ˜Hn(Sn
)
which implies the statement.
Corollary. For any n ≥ 1 and given k ∈ Z there is a map f : Sn
→ Sn
such that
deg f = k.
Proof. For n = 1 put f(z) = zk
where z ∈ S1
⊂ C. Using the computation based on
local degree as above, we get deg f = k. The previous theorem implies that the degree
of Sn−1
f : Sn
→ Sn
is also k.
4.4. Computations of homology of CW-complexes. If we know a CW-structure
of a space X, we can compute its cohomology relatively easily. Consider the sequence
of Abelian groups and its morphisms
(Hn(Xn
, Xn−1
), dn)
where dn is the composition
Hn(Xn
, Xn−1
)
∂n
−−→ Hn(Xn−1
)
jn−1
−−−→ Hn−1(Xn−1
, Xn−2
).
Theorem. Let X be a CW-complex. (Hn(Xn
, Xn−1
), dn) is a chain complex with
homology
HCW
n (X) ∼= Hn(X).
Proof. First, we show how the groups Hk(Xn
, Xn−1
) look like. Put X−1
= ∅ and
X0
/∅ = X0
{∗}. Then
Hk(Xn
, Xn−1
) = ˜Hk(Xn
/Xn−1
) = ˜Hk( Sn
α) = α Z n = k,
0 n = k.
Now we show that
Hk(Xn
) = 0 for k > n.
From the long exact sequence of the pair (Xn
, Xn−1
) we get Hk(Xn
) = Hk(Xn−1
). By
induction Hk
(Xn
) = Hk(X−1
) = 0.
Next we prove that
Hk(Xn
) = Hk(X) for k ≤ n − 1.
From the long exact sequence for the pair (Xn+1
, Xn
) we obtain Hk(Xn
) = Hk(Xn+1
).
By induction Hk(Xn
) = Hk(Xn+m
) for every m ≥ 1. Since the image of each singular
chain lies in some Xn+m
we get Hk(Xn
) = Hk(X).
27
To prove Theorem we will need the following diagram with parts of exact sequences
for the pairs (Xn+1
, Xn
), (Xn
, Xn−1
) and (Xn−1
, Xn−2
).
0
0
&&MMMMMMMMMMMMM Hn(Xn+1
)
OO
Hn(Xn
)
OO
jn
))RRRRRRRRRRRRRR
Hn+1(Xn+1
, Xn
)
∂n+1
OO
dn+1
// Hn(Xn
, Xn−1
)
dn
//
∂n
))SSSSSSSSSSSSSS
Hn−1(Xn−1
, Xn−2
)
Hn−1(Xn−1
)
jn−1
OO
0
OO
From it we get
dndn+1 = jn−1(∂njn)∂n+1 = jn−1(0)∂n+1 = 0.
Further,
Ker dn = Ker ∂n = Im jn
∼= Hn(Xn
)
and
Im dn+1
∼= Im ∂n+1,
since jn−1 and jn are monomorphisms. Finally,
HCW
n (X) =
Ker dn
Im dn+1
∼=
Hn(Xn
)
Im ∂n+1
∼= Hn(Xn+1
) ∼= Hn(X).
Example. Hn(X) = 0 for CW-complexes without cells in dimension n.
Hk(CPn
) =
Z for k ≤ 2n even,
0 in other cases.
4.5. Computation of dn. Let en
α and en−1
β be cells in dimension n and n − 1 of a
CW-complex X, respectively. Since
Hn(Xn
, Xn−1
) =
α
Z, Hn−1(Xn−1
, Xn−2
) =
β
Z,
28
they can be considered as generators of these groups. Let ϕα : ∂Dn
α → Xn−1
be the
attaching map for the cell en
α. Then
dn(en
α) =
β
dαβen−1
β
where dαβ is the degree of the following composition
Sn−1
= ∂Dn
α
ϕα
−→ Xn−1
→ Xn−1
/Xn−2
→ Xn
/(Xn−2
∪
γ=β
en−1
γ ) = Sn−1
.
For the proof we refer to [Hatcher], pages 140 and 141.
Exercise. Compute homology groups of various 2-dimensional surfaces (torus, Klein
bottle, projective plane) using their CW-structure with only one cell in dimension 2.
4.6. Homology of real projective spaces. The real projective space RPn
is formed
by cell e0
, e1
, . . . , en
, one in each dimension from 0 to n. The attaching map for the
cell ek+1
is the projection ϕ : Sk
→ RPk
. So we have to compute the degree of the
composition
f : Sk ϕ
−→ RPk
→ RPk
/RPk−1
= Sk
.
Every point in Sk
has two preimages x1, x2. In a neihbourhood Ui of xi f is a
homeomorphism, hence its local degree deg f|xi = ±1. Since f/U2 is the composition
of the antipodal map with f/U1, the local degrees deg f|x1 and deg f|x1 differs by the
multiple of (−1)k+1
. (See the properties (4) and (6) in 4.2.) According to 4.3
deg f = ±1(1 + (−1)k+1
) =
0 for k + 1 odd,
±2 for k + 1 even.
So we have obtained the chain complex for computation of HCW
∗ (RPn
). The result is
Hk(RPn
) =



Z for k = 0 and k = n odd,
Z2 for k odd , 0 < k < n,
0 in other cases.
4.7. Euler characteristic. Let X be a finite CW-complex. The Euler characteristic
of X is the number
χ(X) =
∞
i=0
(−1)k
rank Hk(X).
Theorem. Let X be a finite CW-complex with ck cells in dimension k. Then
χ(X) =
∞
k=0
(−1)k
ck.
29
Proof. Realize that ck = rank Hk(Xk
, Xk−1
) = rank Ker dk + rank Im dk+1 and that
rank Hk(X) = rank Ker dk − rank Im dk+1. Hence
χ(X) =
∞
k=0
(−1)k
rank Hk(X) =
∞
k=0
(−1)k
(rank Ker dk − rank Im dk+1)
=
∞
k=0
(−1)k
rank Ker dk +
∞
k=0
(−1)k
rank Im dk =
∞
k=0
(−1)k
ck.
Example. 2-dimensional oriented surface of genus g (the number of handles attached
to the 2-sphere) has the Euler characteristic χ(Mg) = 2 − 2g.
2-dimensional nonorientable surface of genus g (the number of M¨obius bands which
replace discs cut out from the 2-sphere) has the Euler characteristic χ(Ng) = 2 − g.
4.8. Lefschetz Fixed Point Theorem. Let G be a finitely generated Abelian group
and h : G → G a homomorphism. The trace tr h is the trace of the homomorphism
Zn ∼= G/ Torsion G → G/ Torsion G ∼= Zn
induced by h.
Let X be a finite CW-complex. The Lefschetz number of a map f : X → X is
L(f) =
∞
i=0
(−1)i
tr Hif.
Notice that L(idX) = χ(X). Similarly as for the Euler characteristic we can prove
Lemma. Let fn : (Cn, dn) → (Cn, dn) be a chain homomorphism. Then
∞
i=0
(−1)i
tr Hif =
∞
i=0
(−1)i
tr fi
whenever the right hand side is defined.
Theorem (Lefschetz Fixed Point Theorem). If X is a finite simplicial complex or its
retract and f : X → X a map with L(f) = 0, then f has a fixed point.
For the proof see [Hatcher], Chapter 2C. Theorem has many consequences.
Corollary A (Brouwer Fixed Point Theorem). Every continuous map f : Dn
→ Dn
has a fixed point.
Proof. The Lefschetz number of f is 1.
In the same way we can prove
Corollary B. If n is even, then every continuous map f : RPn
→ RPn
has a fixed
point.
Corollary C. Let M be a smooth compact manifold in Rn
with nonzero vector field.
Then χ(M) = 0.
30
The converse of this statement is also true.
Outline of the proof. If M has a nonzero vector field, there is a continuous map f :
M → M which is a ”small shift in the direction of the vector field”. Since such a map
has no fixed point, its Lefschetz number has to be zero. Moreover, f is homotopic to
identity and hence
χ(M) = L(idX) = L(f) = 0.
4.9. Homology with coefficients. Let G be an Abelian group. From the singular
chain complex (Cn(X), ∂n) of a space X we make the new chain complex
Cn(X; G) = Cn(X) ⊗ G, ∂G
n = ∂n ⊗ idG .
The homology groups of X with coefficients G are
Hn(X; G) = Hn(C∗(X; G), ∂G
∗ ).
The homology groups defined before are in fact the homology groups with coefficients
Z. The homology groups with coefficients G satisfy all the basic general properties as
the homology groups with integer coefficients with the exception that
Hn(; G) =
0 for n = 0,
G for n = 0.
If the coefficient group G is a field (for instance G = Q or Zp for p a prime), then
homology groups with coefficients G are vector spaces over this field. It often brings
advantages.
The computation of homology with coefficients G can be carried out again using a
CW-complex structure. For instance, we get
Hk(RPn
; Z2) =
Z2 for 0 ≤ k ≤ n,
0 in other cases.
For an application of Z2-coefficients see the proof of the following theorem in [Hatcher],
pages 174–176.
Theorem (Borsuk-Ulam Theorem). Every map f : Sn
→ Sn
satisfying
f(−x) = −f(x)
has an odd degree.
31
5. Singular cohomology
Cohomology forms a dual notion to homology. To every topological space we assign
a graded group H∗
(X) equipped with a ring structure given by a product ∪ : Hi
(X)×
Hj
(X) → Hi+j
(X). In this section we give basic definitions and properties of singular
cohomology groups which are very similar to those of homology groups.
5.1. Cochain complexes. A cochain complex (C, δ) is a sequence of Abelian groups
(or modules over a ring) and their homomorphisms indexed by integers
. . .
δn−2
−−−−→ Cn−1 δn−1
−−−−→ Cn δn
−−→ Cn+1 δn+1
−−−−→ . . .
such that
δn
δn−1
= 0.
δn
is called a coboundary operator. A cochain homomorphism of cochain complexes
(C, δC) and (D, δD) is a sequence of homomorphisms of Abelian groups (or modules
over a ring) fn
: Cn
→ Dn
which commute with the coboundary operators
δn
Dfn = fn+1
δn
C.
5.2. Cohomology of cochain complexes. The n-th cohomology group of a cochain
complex (C, δ) is the group
Hn
(C) =
Ker δn
Im δn−1
.
The elements of Ker δn
= Zn
are called cocycles of dimension n and the elements of
Im δn−1
= Bn
are called coboundaries (of dimension n). If a cochain complex is exact,
then its cohomology groups are trivial.
The component fn
of the cochain homomorphism f : (C, δC) → (D, δD) maps
cocycles into cocycles and coboundaries into coboundaries. It enables us to define
Hn
(f) : Hn
(C) → Hn
(D)
by the prescription Hn
(f)[c] = [fn
(c)] where [c] ∈ Hn
(C) and [fn
(c)] ∈ Hn
(D) are
classes represented by the elements c ∈ Zn
(C) and fn
(c) ∈ Zn
(D), respectively.
5.3. Long exact sequence in cohomology. A sequence of cochain homomorphisms
· · · → A
f
−−→ B
g
−−→ C → . . .
is exact if for every n ∈ Z
· · · → An fn
−−→ Bn
gn
−−→ Cn
→ . . .
is an exact sequence of Abelian groups. Similarly as for homology groups we can prove
Theorem. Let 0 → A
i
−→ B
j
−−→ C → 0 be a short exact sequence of cochain complexes.
Then there is a so called connecting homomorphism δ∗
: Hn
(C) → Hn+1
(A)
32
such that the sequence
. . .
δ∗
−−→ Hn
(A)
Hn(i)
−−−→ Hn
(B)
Hn(j)
−−−→ Hn
(C)
δ∗
−−→ Hn+1
(A)
Hn+1(i)
−−−−→ . . .
is exact.
5.4. Cochain homotopy. Let f, g : C → D be two cochain homomorphisms. We
say that they are cochain homotopic if there are homomorphisms sn
: Cn
→ Dn−1
such that
δn−1
D sn
+ sn+1
δn
C = fn
− gn
for all n.
The relation to be cochain homotopic is an equivalence. The sequence of maps sn
is
called a cochain homotopy. Similarly as for homology we have
Theorem. If two cochain homomorphism f, g : C → D are cochain homotopic, then
Hn
(f) = Hn
(g).
5.5. Singular cohomology groups of a pair. Consider a pair of topological spaces
(X, A), an inclusion i : A → X and an Abelian group G. Let
C(X, A) = (Cn(X)/Cn(A), ∂n)
be the singular chain complex of the pair (X, A). The singular cochain complex
(C(X, A; G), δ) for the pair (X, A) is defined as
Cn
(X, A; G) = Hom (Cn(X, A), G) ∼= {h ∈ Hom(Cn(X), G); h|Cn(A) = 0}
= Ker i∗
: Hom(Cn(X), G) −→ Hom(Cn(A), G).
and
δn
(h) = h ◦ ∂n+1 for h ∈ Hom(Cn(X, A), G).
The n-th cohomology group of the pair (X, A) with coefficients in the group G is the
n-th cohomology group of this cochain complex
Hn
(X, A; G) = Hn
(C(X, A; G), δ).
We write Hn
(X; G) for Hn
(X, ∅; G). A map f : (X, A) → (Y, B) induces the cochain
homomorphism Cn
(f) : Cn
(Y ; G) → Cn
(X; G) by
Cn
(f)(h) = h ◦ Cn(f)
which restricts to a cochain homomorphism Cn
(Y, B; G) → Cn
(X, A; G) since f(A) ⊆
B. In cohomology it induces the homomorphism
f∗
= Hn
(f) : Hn
(Y, B) → Hn
(X, A).
Moreover, Hn
(id(X,A)) = idHn(X,A;G) and Hn
(fg) = Hn
(g)Hn
(f). We can conclude
that Hn
is a contravariant functor (cofunctor) from the category Top2
into the category
AG of Abelian groups.
33
5.6. Long exact sequence for singular cohomology. Consider inclusions of spaces
i : A → X, i : B → Y and maps j : (X, ∅) → (X, A), j : (Y, ∅) → (Y, B) induced
by idX and idY , respectively. Let f : (X, A) → (Y, B) be a map. Then there are
connecting homomorphisms δ∗
X and δ∗
Y such that the following diagram
. . .
δ∗
X
// Hn
(X, A; G)
j∗
// Hn
(X; G)
i∗
// Hn
(A; G)
δ∗
X
// Hn+1
(X, A; G)
j∗
// . . .
. . .
δ∗
Y
// Hn
(X, B; G)
j ∗
//
f∗
OO
Hn
(Y ; G)
i ∗
//
f∗
OO
Hn
(B; G)
δ∗
Y
//
(f/B)∗
OO
Hn+1
(Y, B; G)
j ∗
//
f∗
OO
. . .
commutes and its horizontal sequences are exact.
The proof follows from Theorem 5.3 using the fact that
0 → Cn
(X, A; G)
Cn(j)
−−−→ Cn
(X; G)
Cn(i)
−−−→ Cn
(A; G) → 0
is a short exact sequence of cochain complexes as it follows directly from the definition
of Cn
(X, A; G).
Remark A. Consider the functor I : Top2
→ Top2
which assigns to every pair (X, A)
the pair (A, ∅). The commutativity of the last square in the diagram above means
that δ∗
is a natural transformation of contravariant functors Hn
◦ I and Hn+1
defined
on Top2
.
Remark B. It is useful to realize how δ∗
: Hn
(A; G) → Hn+1
(X, A; G) looks like.
Every element of Hn
(A; G) is represented by a cochain q ∈ Hom(Cn(A); G) with a zero
coboundary δq ∈ Hom(Cn+1(A); G). Extend q to Q ∈ Hom(Cn(X); G) in arbitrary
way. Then δQ ∈ Hom(Cn+1(X), G) restricted to Cn+1(A) is equal to δq = 0. Hence it
lies in Hom(Cn+1(X, A); G) and from the definition in 5.3 we have
δ∗
[q] = [δQ].
5.7. Homotopy invariance. If two maps f, g : (X, A) → (Y, B) are homotopic, then
they induce the same homomorphisms
f∗
= g∗
: Hn
(Y, B; G) → Hn(X, A; G).
Proof. We already know that the homotopy between f and g induces a chain homotopy
s∗ between C∗(f) and C∗(g). Then we can define a cochain homotopy between C∗
(f)
and C∗
(g) as
sn
(h) = h ◦ sn−1 for h ∈ Hom(Cn(Y ); G)
and use Theorem 5.4.
Corollary. If X and Y are homotopy equivalent spaces, then
Hn
(X) ∼= Hn
(Y ).
5.8. Excision Theorem. Similarly as for singular homology groups there are two
equivalent versions of this theorem.
34
Theorem A (Excision Theorem, 1st version). Consider spaces C ⊆ A ⊆ X and
suppose that ¯C ⊆ int A. Then the inclusion
i : (X − C, A − C) → (X, A)
induces the isomorphism
i∗
: Hn
(X, A; G)
∼=
−→ Hn
(X − C, A − C; G).
Theorem B (Excision Theorem, 2nd version). Consider two subspaces A and B of a
space X. Suppose that X = int A ∪ int B. Then the inclusion
i : (B, A ∩ B) → (X, A)
induces the isomorphism
i∗
: Hn
(X, A; G)
∼=
−→ Hn
(B, A ∩ B; G).
The proof of Excision Theorem for singular cohomology follows from the proof of
the homology version.
5.9. Cohomology of finite disjoint union. Let X = k
α=1 Xα be a disjoint union.
Then
Hn
(X; G) =
k
α=1
Hn
(Xα).
The statement is not generally true for infinite unions.
5.10. Reduced cohomology groups. For every space X = ∅ we define the augmented
cochain complex ( ˜C∗
(X; G), ˜δ) as follows
˜Cn
(X; G) = Hom( ˜Cn(X); G)
with ˜δn
h = h◦ ˜∂n+1 for h ∈ Hom( ˜Cn(X); G). See 3.14. The reduced cohomology groups
˜Hn(X; G) with coefficients in G are the cohomology groups of the augmented cochain
complex. From the definition it is clear that
˜Hn
(X; G) = Hn
(X; G) for n = 0
and
˜Hn
(∗; G) = 0 for all n.
For pairs of spaces we define ˜Hn
(X, A; G) = Hn
(X, A; G) for all n. Then theorems
on long exact sequence, homotopy invariance and excision hold for reduced cohomology
groups as well.
Considering a space X with base point ∗ and applying the long exact sequence for
the pair (X, ∗), we get that for all n
˜Hn
(X; G) = ˜Hn
(X, ∗; G) = Hn
(X, ∗; G).
Using this equality and the long exact sequence for unreduced cohomology we get that
H0
(X; G) ∼= H0
(X, ∗; G) ⊕ H0
(∗; G) ∼= ˜H0
(X) ⊕ G.
35
Analogously as for homology groups we have
Lemma. Let (X, A) be a pair of CW-complexes. Then
˜Hn
(X/A; G) = Hn
(X, A; G)
and we have the long exact sequence
· · · → ˜Hn
(X/A; G) → ˜Hn
(X; G) → ˜Hn
(A; G) → ˜Hn+1
(X/A; G) → . . .
5.11. The long exact sequence of a triple. Consider a triple (X, B, A), A ⊆
B ⊆ X. Denote i : (B, A) → (X, A) and j : (X, A) → (X, B) maps induced by the
inclusion B → X and idX, respectively. Analogously as for homology one can derive
the long exact sequence of the triple (X, B, A)
. . .
δ∗
−−→ Hn
(X, B; G)
j∗
−−→ Hn
(X, A; G)
i∗
−−→ Hn
(B, A; G)
δ∗
−−→ Hn+1
(X, B; G)
j∗
−−→ . . .
5.12. Singular cohomology groups of spheres. Considering the long exact sequence
of the triple (∆n
, δ∆n
, V = δ∆n
− ∆n−1
): we get that
Hi
(∆n
, ∂∆n
; G) =
G for i = n,
0 for i = n.
The pair (Dn
, Sn−1
) is homeomorphic to (∆n
, ∂∆n
). Hence it has the same cohomology
groups. Using the long exact sequence for this pair we obtain
˜Hi
(Sn
; G) = Hi+1
(Dn+1
, Sn
) =
0 for i = n,
G for i = n.
5.13. Mayer-Vietoris exact sequence. Denote inclusions A∩B → A, A∩B → B,
A → X, B → X by iA, iB, jA, jB, respectively. Let C → A, D → B and suppose
that X = int A ∪ int B, Y = int C ∪ int D. Then there is the long exact sequence
. . .
δ∗
−−→ Hn
(X, Y ; G)
(j∗
A,j∗
B)
−−−−−→ Hn
(A, C; G) ⊕ Hn
(B, D; G)
i∗
A−i∗
B
−−−−−→ Hn(A ∩ B, C ∩ D; G)
δ∗
−−→ Hn+1
(X, Y ; G) −→ . . .
Proof. The coverings U = {A, B} and V = {C, D} satisfy conditions of Lemma 3.12.
The sequence of chain complexes
0 −→ Cn(A ∩ B, C ∩ D)
i
−→ Cn(A, C) ⊕ Cn(B; D)
j
−→ CU,V
n (X, Y ) −→ 0
where i(x) = (x, x) and j(x, y) = x − y is exact. Applying Hom(−, G) we get a new
short exact sequence of cochain complexes
0 −→ Cn
U,V(X, Y ; G)
j∗
−→ Cn
(A, C; G) ⊕ Cn
(B, D; G)
i∗
−→ Cn
(A ∩ B, C ∩ D; G) −→ 0
and it induces a long exact sequence. Using Lemma 3.12 we get that Hn
(CU,V(X, Y ; G)) =
Hn
(X, Y ; G), which completes the proof.
36
5.14. Computations of cohomology of CW-complexes. If we know a CWstructure
of a space X, we can compute its cohomology in the same way as homology.
Consider the chain complex from Section 4
(Hn(Xn
, Xn−1
), dn).
Theorem. Let X be a CW-complex. The n-th cohomology group of the cochain com-
plex
(Hom(Hn(Xn
, Xn−1
; G), dn
) dn
(h) = h ◦ dn
is isomorphic to the n-th singular cohomology group Hn
(X; G).
Exercise A. After reading the next section try to prove the theorem above using the
results and proofs from Section 4.
Exercise B. Compute singular cohomology of real and complex projective spaces
with coefficients Z and Z2.
37
6. More homological algebra
In this section we will deal with algebraic constructions leading to the definitions
of homology and cohomology groups with coefficients given in the previous sections.
At the end we use introduced notions to state and prove so called universal coefficient
theorems for singular homology and cohomology groups.
6.1. Functors and cofunctors. Let A and B be two categories. A functor t : A → B
assigns to every object x in A an object t(x) in B and to every morphism f : x → y in
A a morphism t(f) : t(x) → t(y) such that t(idx) = idt(x) and t(fg) = t(f)t(g).
A contravariant functor or briefly cofunctor t : A → B assigns to every object x in A
an object t(x) in B and to every morphism f : x → y in A a morphism t(f) : t(y) → t(x)
in B such that t(idx) = idt(x) and t(fg) = t(g)t(f).
Let R be a commutative ring with a unit element. The category of R-modules
and their homomorphisms will be denoted by R-Mod. R-GMod will be used for the
category of graded R-modules, R-Ch and R-CoCh will stand for the categories of chain
complexes and the category of cochain complexes of R-modules, respectively. For
R = Z the previous categories are Abelian groups Ab, graded Abelian groups GAb,
chain complexes of Abelian groups Ch and cochain complexes of Abelian groups CoCh,
respectively.
Homology H is a functor from the category R-Ch to the category R-GMod. Let t
be a functor from R-Mod to R-Mod which induces a functor t : Ch → Ch, and let s be
a cofunctor from R-Mod to R-Mod, which induces a cofunctor from Ch to CoCh. The
aim of this section is to say something about the functor H ◦t and the cofunctor H ◦s.
Model examples of such functors will be t(−) = − ⊗R M and s(−) = HomR(−, M)
for a fixed R-module M. We have already used these functors when we have defined
homology and cohomology groups with coefficients.
6.2. Tensor product. The tensor product A ⊗R B of two R-modules A and B is the
quotient of the free R-module over A × B and the ideal generated by the elements of
the form
r(a, b)−(ra, b), r(a, b)−(a, rb), (a1 +a2, b)−(a1, b)−(a2, b), (a, b1 +b2)−(a, b1)−(a, b2)
where a, a1, a2 ∈ A, b, b1, b2 ∈ B, r ∈ R. The class of equivalence of the element (a, b)
in A ⊗R B is denoted by a ⊗ b. The map ϕ : A × B → A ⊗R B, ϕ(a, b) = a ⊗ b is
bilinear and has the following universal property:
Whenever an R-module C and a bilinear map ψ : A × B → C are given, there is
just one R-modul homomorphism Ψ : A ⊗R B → C such that the diagram
A × B
ψ
//
ϕ

C
A ⊗R B
Ψ
;;vvvvvvvvv
commutes. This property determines the tensor product uniquely up to isomorphism.
38
If f : A → C and g : B → D are homomorphisms of R-modules then (a, b) →
f(a) ⊗ g(b) is a bilinear map and the universal property above ensures the existence
and uniqueness of an R-homomorphism f ⊗ g : A ⊗R B → C ⊗R D with the property
(f ⊗ g)(a ⊗ b) = f(a) ⊗ g(b).
Homomorphisms between R-modules form an R-module denoted by HomR(A, B).
If R = Z, we will denote the tensor product of Abelian groups A and B without the
subindex Z, i.e. A ⊗ B, and similarly, the group of homomorfisms from A to B will
be denoted by Hom(A, B).
Exercise. Prove from the definition that
Z ⊗ Z = Z, Z ⊗ Zn = Zn, Zn ⊗ Zm = Zd(n,m), Zn ⊗ Q = 0
where d(m, n) is the greatest common divisor of n and m. Further compute
Hom(Z, Z), Hom(Z, Zn), Hom(Zn, Z), Hom(Zn, Zm).
6.3. Additive functors and cofunctors. A functor (or a cofunctor) t : Mod → Mod
is called additive if
t(α + β) = t(α) + t(β)
for all α, β ∈ HomR(A, B). Additive functors and cofunctors have the following prop-
erties.
(1) t(0) = 0 for any zero homomorphism.
(2) t(A ⊕ B) = t(A) ⊕ t(B)
(3) Every additive functor (cofunctor) converts short exact sequences which split
into short exact sequences which again split.
(4) Every additive functor (cofunctor) can be extended to a functor Ch → Ch
(cofunctor Ch → CoCh) which preserves chain homotopies (converts chain homotopies
to cochain homotopies).
Proof of (2) and (3). Consider a short exact sequence
0 → A
i
−→ B
j
−→ C → 0
which splits, i. e. there are homomorphisms p : B → A, q : C → B such that pi = idA,
jq = idC, ip + qj = idB. See 3.1. Applying an additive functor t we get a splitting
short exact sequence described by homomorphisms t(i), t(j), t(p), t(q).
6.4. Exact functors and cofunctors. An additive functor (or an additive cofunctor)
t : Mod → Mod is called exact if it preserves short exact sequences.
Example. The functor t(−) = − ⊗ Z2 and the cofunctor s(−) = Hom(−, Z) are
additive but not exact. To show it apply them on the short exact sequence
0 → Z
2×
−→ Z → Z2 → 0.
On the other hand, the functor t(−) = − ⊗ Q from Ab to Ab is exact.
39
Lemma. Let (C, ∂) be a chain complex and let t : Mod → Mod be an exact functor.
Then
Hn(tC, t∂) = tHn(C, ∂).
Consequently, t converts all exact sequences into exact sequences.
Proof. Since t preserves short exact sequences, it preserves kernels, images and factors.
So we get
Hn(tC) =
Ker t∂n
Im t∂n+1
=
t(Ker ∂n)
t(Im ∂n+1)
= t
Ker ∂n
Im ∂n+1
= t(Hn(C)).
6.5. Right exact functors. An additive functor t : Mod → Mod is called right exact
if it converts any exact sequence
A
i
−→ B
j
−→ C → 0
into an exact sequence
tA
t(i)
−−→ tB
t(j)
−−→ tC → 0.
Theorem. Consider an R-module M. The functor t(−) = − ⊗R M from Mod to Ab
is right exact.
Proof. The exact sequence A
i
−→ B
j
−→ C → 0 is converted into the sequence
A ⊗R M
i⊗idM
−−−→ B ⊗R M
j⊗idM
−−−−→ C ⊗R M → 0.
It is clear that j ⊗idM is an epimorphism. According to the lemma below Ker(j ⊗idM )
is generated by elements b ⊗ m where b ∈ Ker j = Im i. Hence, Ker(j ⊗ idM ) =
Im(i ⊗ idM ).
Lemma. If α : A → A and β : B → B are epimorphisms, then Ker(α ⊗ β) is
generated by elements a ⊗ b where a ∈ Ker α or b ∈ Ker β.
For the proof see [Spanier], Chapter 5, Lemma 1.5.
6.6. Left exact cofunctors. An additive cofunctor t : Mod → Mod is called left
exact if it converts any exact sequence
A
i
−→ B
j
−→ C → 0
into an exact sequence
O → tC
t(j)
−−→ tB
t(i)
−−→ tA.
Theorem. Consider an R-module M. The cofunctor t(−) = HomR(−, M) from Mod
to Mod is left exact.
The proof is not difficult and is left as an exercise.
40
6.7. Projective modules. An R-modul is called projective if for any epimorphism
p : A → B and any homomorphism f : P → B there is F : P → A such that the
diagram
A
p

P
F
??~~~~~~~~
f
// B

0
commutes. Every free R-module is projective.
6.8. Projective resolution. A projective resolution of an R-module A is a chain
complex (P, ε), Pi = 0 for i < 0 and a homomorphism α : P0 → A such that the
sequence
→ Pi
εi
−→ Pi−1 → · · · → P1
ε1
−→ P0
α
−→ A → 0
is exact. It means that
Hi(P, ε) =
0 for i = 0,
P0/ Im ε1 = P0/ Ker α ∼= A for i = 0.
If all Pi are free modules, the resolution is called free.
Lemma A. To every module there is a free resolution.
Proof. For module A denote F(A) a free module over A and π : F(A) → A a canonical
projection. Then the free resolution of A is constructed in the following way
P2 = F Ker ε1
//
ε2
--
Ker ε1
// P1 = F(Ker π) //
ε1
,,
Ker π // P0 = F(A) π
// A
Lemma B. Every Abelian group A has the projective resolution
0 → Ker π → F(A)
π
−→ A → 0.
Proof. Ker π as a subgroup of free Abelian group F(A) is free.
Theorem. Consider a homomorphism of R-modules ϕ : A → A . Let (Pn, εn) and
(Pn, εn) be projective resolutions of A and A , respectively. Then there is a chain
homomorphism ϕn : (P, ε) → (P , ε ) such that the diagram
. . .
ε3
// P2
ε2
//
ϕ2

P1
ε1
//
ϕ1

P0
α //
ϕ0

A //
ϕ

0
. . .
ε3
// P2
ε2
// P1
ε1
// P0
α // A // 0
commutes. Moreover, any two such chain homomorphism (P, ε) → (P , ε ) are chain
homotopic.
41
Proof of the first part. α is an epimorphism and P0 is projective. Hence there is ϕ0 :
P0 → P0 such that the first square on the right side commutes.
Since α (ϕ0ε1) = ϕ(αε1) = ϕ ◦ 0 = 0, we get that
Im(ϕ0ε) ⊆ Ker α = Im ε1.
ε1 : P1 → Im ε1 is an epimorhism and P1 is projective. Hence there is ϕ1 : P1 → P1
such that the second square in the diagram commutes.
The proof of the rest of the first part proceeds in the same way by induction.
Proof of the second part. Let ϕ∗ and ϕ∗ be two chain homomorphisms making the
diagram above commutative. Since α (ϕ0 − ϕ0) = α(ϕ − ϕ) = 0, we have
Im(ϕ0 − ϕ0) ⊆ Ker α = Im ε1.
Therefore there exists s0 : P0 → P1 such that ε1s0 = ϕ0 − ϕ0.
Next, ε1(ϕ1 − ϕ1 − s0ε1) = ε1(ϕ1 − ϕ1) − ε1s0ε1 = (ϕ0 − ϕ0)ε1 − (ϕ0 − ϕ0)ε1 = 0,
hence
Im(ϕ0 − ϕ0 − s0ε1) ⊆ Ker ε1 = Im ε2,
and consequently, there is s1 : P1 → P2 such that
ε2s1 = ϕ1 − ϕ1 − s0ε1.
The rest proceeds by induction in the same way.
6.9. Derived functors. Consider a right exact functor t : Mod → Mod and a homomorphism
of R-modules ϕ : A → A . Let (P, ε) and (P , ε ) be projective resolutions
of A and A , respectively, and let ϕ∗ : (P, ε) → (P , ε ) be a chain homomorphism
induced by ϕ. The derived functors ti : Mod → Mod of the functor t are defined
tiA = Hi(tP, tε)
tiϕ = Hi(tϕ).
The functor t0 is equal to t since
t0A = tP0/ Im tε1 = tP0/ Ker tα ∼= tA.
Using the previous theorem we can easily show that the definition does not depend on
the choice of projective resolutions and a chain homomorphism ϕ∗.
Definition. The i-th derived functors of the functor t(−) = − ⊗R M is denoted
TorR
i (−, M).
If R = Z, the index Z in the notation will be omitted.
Example. Let R = Z. Any Abelian group A has a free resolution with Pi = 0 for
i ≥ 2. Hence
Tori(A, B) = 0 for i ≥ 2.
Hence we will omit the index 1 in Tor1(A, B). We have
(1) Tor(A, B) = 0 for any free Abelian group A.
(2) Tor(A, B) = 0 for any free Abelian group B.
42
(3) Tor(Zm, Zn) = Zd(m,n) where d(m, n) is the greatest common divisor of m and
n.
(4) Tor(−, B) is an additive functor.
The proof based on the definition is not difficult and is left to the reader as an exercise.
6.10. Derived cofunctors. Consider a left exact cofunctor t : Mod → Mod and a
homomorphism of R-modules ϕ : A → A . Let (P, ε) and (P , ε ) be projective resolutions
of A and A , respectively, and let ϕ∗ : (P, ε) → (P , ε ) be a chain homomorphism
induced by ϕ. The derived cofunctors ti
: Mod → Mod of the functor t are defined
ti
A = Hi
(tP, tε)
ti
ϕ = Hi
(tϕ).
The functor t0
is equal to t since
t0A = Ker tε1 = Im tα = tA.
Using Theorem 6.8 we can easily show that the definition does not depend on the
choice of projective resolutions and a chain homomorphism ϕ∗.
Definition. The i-th derived functors of the functor t(−) = HomR(−, M) is denoted
Exti
R(−, M).
If R = Z, the index Z in the notation will be omitted.
Example. Let R = Z. Since every Abelian group A has a free resolution with Pi = 0
for i ≥ 2,
Exti
(A, B) = 0 for i ≥ 2.
Hence we will write Ext(A, B) for Ext1
(A, B). We have
(1) Ext(A, B) = 0 for any free Abelian group A.
(2) Ext(Zn, Z) = Zn.
(3) Ext(Zm, Zn) = Zd(m,n) where d(m, n) is the greatest common divisor of m and
n.
(4) Ext(−, B) is an additive cofunctor.
The proof is the application of the definition and it is again left to the reader.
6.11. Universal coefficient theorems. In this paragraph we first express the cohomology
groups Hn
(X; G) with the aid of functors Hom and Ext using the homology
groups H∗(X).
Theorem A. If a free chain complex C of Abelian groups has homology groups Hn(C),
then the cohomology groups Hn
(C; G) of the cochain complex Cn
= Hom(Cn, G) are
determined by the following split short exact sequence
0 → Ext(Hn−1(C), G) → Hn
(C; G)
h
−→ Hom(Hn(C), G) → 0
where h[f]([c]) = f(c) for all cycles c ∈ Cn and all cocycles f ∈ Hom(Cn; G).
43
Remark. The exact sequence is natural but the splitting not. In this case the naturality
means that for every chain homomorphism f : C → D we have commutative
diagram
0 // Ext(Hn−1(C), G) //
Ext(Hn−1f,idG)

Hn
(C; G)
hC
//
Hnf

Hom(Hn(C), G) //
Hom(Hnf,idG)

0
0 // Ext(Hn−1(D), G) // Hn
(D; G)
hD
// Hom(Hn(D), G) // 0
Proof. The free chain complex (Cn, ∂) determines two other chain complexes, the chain
complex of cycles (Zn, 0) and the chain complex of boundaries (Bn, 0). We have the
short exact sequence of these chain complexes
0 → Zn
i
−→ Cn
∂
−→ Bn−1 → 0.
Since Bn−1 is a subgroup of the free Abelian group Cn−1, it is also free and the exact
sequence splits. Since the functor Hom(−, G) is additive, it converts this sequence into
the short exact sequence of cochain complexes
0 → Hom(Bn−1, G)
δ
−→ Hom(Cn, G)
i∗
−→ Hom(Zn, G) → 0.
As in 5.6 we obtain the long exact sequence of cohomology groups of the given cochain
complexes
→ Hom(Zn−1, G) → Hom(Bn−1, G) → Hn
(C; G)
i∗
−→ Hom(Zn, G)
δ∗
−→ Hom(Bn, G) →
Next, one has to realize how the connecting homomorphism δ∗
in this exact sequence
looks like using its definition and the special form of the short exact sequence. The
conclusion is that δ∗
= j∗
where j : Bn → Zn is an iclusion. Now we can reduce the
long exact sequence to the short one
0 →
Hom(Bn−1, G)
Im j∗
−→ Hn
(C; G)
i∗
−→ Ker j∗
→ 0.
We determine Ker j∗
and Hom(Bn−1, G)/ Im j∗
. Consider the short exact sequence
0 → Bn
j
−→ Zn → Hn(C) → 0.
It is a free resolution of Hn(C). Applying the cofunctor Hom(−, G) we get the cochain
complex
0 → Hom(Zn, G)
j∗
−→ Hom(Bn, G) → 0 → 0 → . . .
from which we can easily compute that
Hom(Hn(C), G) = Ker j∗
, Ext(Hn(C), G) =
Hom(Bn−1, G)
Im j∗
.
This completes the proof of exactness.
We will find a splitting r : Hom(Hn(C); G) → Hn
(C∗
; G). Let g ∈ Hom(Hn(C), G).
We can define f ∈ Hom(Cn, G) such that on cycles c ∈ Zn
f(c) = g([c])
where [c] ∈ Hn(C). f is a cocycle, hence [f] ∈ Hn
(C; G) and h[f]([c]) = f(c) =
g([c]).
44
In a very similar way one can compute homology groups with coefficients in G using
the tensor product H∗(C) ⊗ G and Tor(H∗(C), G).
Theorem B. If a free chain complex C has homology groups Hn(C), then the homology
groups Hn(C∗; G) of the chain complex Cn ⊗ G are determined by the split short exact
sequence
0 → Hn(C) ⊗ G
l
−→ Hn(C; G) → Tor(Hn−1(C), G) → 0
where l([c] ⊗ g) = [c ⊗ g] for c ∈ Zn(C), g ∈ G.
6.12. Exercise. Compute cohomology of real projective spaces with Z and Z2 coefficients
using the universal coefficient theorem for cohomology.
Exercise. Using again the universal coefficient theorem for cohomology and Theorem
4.4 prove that that for a given CW-complex X the cohomology of the cochain complex
(Hn
(Xn
, Xn−1
; G), dn
)
where dn
is the composition
Hn
(Xn
, Xn−1
; G)
j∗
n
−→ Hn
(Xn
; G)
δ∗
−→ Hn+1
(Xn+1
, Xn
; G)
is isomorphic to H∗
(X; G). See also Theorem 5.14.
45
7. Products in cohomology
An internal product in cohomology brings a further algebraic structure. The contravariant
functor H∗
becomes a cofunctor into graded rings. It enables us to obtain
more information on topological spaces and homotopy classes of maps. In this section
we will define an internal product – called the cup product and a closely related
external product – called the cross product.
7.1. Cup product. Let R be a commutative ring with a unite and let X be a
space. For two cochains ϕ ∈ Ck
(X; R) and ψ ∈ Cl
(X; R) we define their cup product
ϕ ∪ ψ ∈ Ck+l
(X; R)
(ϕ ∪ ψ)(σ) = ϕ(σ/[v0, v1, . . . , vk]) · ψ(σ/[vk, vk+1, . . . , vk+l])
for any singular simplex σ : ∆k+l
→ X. The notation σ/[v0, v1, . . . , vk] and σ/[vk, vk+1,
. . . , vk+l] stands for σ composed with inclusions of the standard simplices ∆k
and ∆l
into the indicated faces of the standard simplex ∆k+l
, respectively. The coboundary
operator δ behaves on the cup products of cochains as graded derivation as shown in
the following
Lemma.
δ(ϕ ∪ ψ) = δϕ ∪ ψ + (−1)k
ϕ ∪ δψ.
Proof. For σ ∈ Ck+l+1(X) we get
(δϕ ∪ ψ)(σ) + (−1)k
(ϕ ∪ δψ)(σ) = δϕ(σ/[v0, v1, . . . , vk+1])ψ(σ/[vk+1, . . . , vk+l+1])
+ (−1)k
ϕ(σ[v0, v1, . . . , vk])δψ(σ/[vk, . . . , vk+l+1])
=
k+1
i=0
(−1)i
ϕ(σ/[v0, . . . , ˆvi, . . . , vk+1])(ψ(σ/[vk+1, . . . , vk+l+1]))
+ (−1)k
k+l+1
j=k
(−1)j−k
ϕ(σ/[v0, . . . , vk])ψ(σ/[vk, . . . , ˆvj, . . . , vk+l+1])
=
k+l+1
i=0
(−1)i
(ϕ ∪ ψ)(σ/[v0, . . . , ˆvi, . . . , vk+l+1]) = δ(ϕ ∪ ψ)(σ).
Lemma implies that
(1) If ϕ and ψ are cocycles, then ϕ ∪ ψ is a cocycle.
(2) If one of the cochains ϕ and ψ is a coboundary, then ϕ ∪ ψ is a coboundary.
It enables us to define the cup product
∪ : Hk
(X; R) × Hl
(X; R) → Hk+l
(X; R)
by the prescription
[ϕ] ∪ [ψ] = [ϕ ∪ ψ]
46
for cocycles ϕ and ψ. Since ∪ is an R-bilinear map on Hk
(X; R)×Hl
(X; R), it can be
considered as an R-linear map on the tensor product Hk
(X; R) ⊗R Hl
(X; R). Given
a pair of spaces (X, A) we can define the cup product as a linear map
∪ :Hk
(X, A; R) ⊗R Hl
(X; R) → Hk+l
(X, A; R),
∪ :Hk
(X; R) ⊗R Hl
(X, A; R) → Hk+l
(X, A; R),
∪ :Hk
(X, A; R) ⊗R Hl
(X, A; R) → Hk+l
(X, A; R).
Moreover, if A and B are open in X or A and B are subcomplexes of CW-complex
X, one can define
∪ : Hk
(X, A; R) ⊗R Hl
(X, B; R) → Hk+l
(X, A ∪ B; R).
Exercise. Prove that the previous definitions of cup product for pairs of spaces are
correct. For the last case you need Lemma 3.12.
Remark. In the same way as the singular cohomology groups and the cup product
have been defined using the singular chain complexes, we can introduce simplicial
cohomology groups for ∆-complexes and a cup product in these groups.
7.2. Properties of the cup product are following:
(1) The cup product is associative.
(2) If X = ∅, there is an element 1 ∈ H0
(X; R) such that for all α ∈ Hk
(X, A; R)
1 ∪ α = α ∪ 1 = α.
(3) For all α ∈ Hk
(X, A; R) and β ∈ Hl
(X, A; R)
α ∪ β = (−1)kl
β ∪ α,
i. e. the cup product is graded commutative.
(4) Naturality of the cup product. For every map f : (X, A) → (Y, B) and any
α ∈ Hk
(Y, B; R), β ∈ Hl
(Y, B; R) we have
f∗
(α ∪ β) = f∗
(α) ∪ f∗
(β).
Remark. Properties (1) – (3) mean that H∗
(X, A; R) = ∞
i=0 Hi
(X, A; R) with the
cup product is not only a graded group but also a graded ring and that H∗
(X; R) is
even a graded ring with a unit if X = ∅. Property (4) says that f : (X, A) → (Y, B)
induces a ring homomorphism f∗
: H∗
(Y, B; R) → H∗
(X, A; R).
Proof. To prove properties (1), (2) and (4) is easy and left to the reader as an exercise.
To prove property (3) is more difficult. We refer to [Hatcher], Theorem 3.14, pages
215 – 217 for geometrically motivated proof. Another approach is outlined later in
7.8.
7.3. Cross product. Consider spaces X and Y and projections p1 : X ×Y → X and
p2 : X × Y → Y . We will define the cross product or external product. The absolute
47
and relative forms are the linear maps
µ : Hk
(X, R) ⊗ Hl
(Y ; R) → Hk+l
(X × Y ; R),
µ : Hk
(X, A; R) ⊗ Hl
(Y, B; R) → Hk+l
(X × Y, A × Y ∪ X × B; R)
given by
µ(α ⊗ β) = p∗
1(α) ∪ p∗
2(β).
For the relative form of the cross product we suppose that A and B are open in X and
Y , or that A and B are subcomplexes of X and Y , respectively. (See the definition of
the cup product.) The name cross product comes from the notation since µ(α ⊗ β) is
often written as α × β.
Exercise. Let ∆ : X → X × X be the diagonal ∆(x) = (x, x). Show that for
α, β ∈ H∗
(X; R)
α ∪ β = ∆∗
µ(α ⊗ β) .
7.4. Tensor product of graded rings. Let A∗
= ∞
n=0 An
and B∗
= ∞
n=0 Bn
be graded rings. Then the tensor product of graded rings A∗
⊗ B∗
is the graded ring
C∗
= ∞
n=0 Cn
where
Cn
=
i+j=n
Ai
⊗ Bj
with the multiplication given by
(a1 ⊗ b1) · (a2 ⊗ b2) = (−1)|b1|·|a2|
(a1 · a2) ⊗ (b1 · b2).
Here |b1| is the degree of b1 ∈ B∗
, i.e. b1 ∈ B|b1|
. If A∗
and B∗
are graded commutative,
so is A∗
⊗ B∗
.
Lemma. The cross product
µ : Hk
(X, A; R) ⊗ Hl
(Y, B; R) → Hk+l
(X × Y, A × Y ∪ X × B; R)
is a homomorphism of graded rings.
Proof. Using the definitions of the cup and cross products and their properties we have
µ (a × b) · (c × d ) = (−1)|b|·|c|
µ (a ∪ c) ⊗ (b ∪ d) = (−1)|b|·|c|
p∗
1(a ∪ c) ∪ p∗
2(b ∪ d)
= (−1)|b|·|c|
p∗
1(a) ∪ p∗
1(c) ∪ p∗
2(b) ∪ p∗
2(d)
= p∗
1(a) ∪ p∗
2(b) ∪ p∗
1(c) ∪ p∗
2(d) = µ(a ⊗ b) ∪ µ(c ⊗ d).
7.5. K¨unneth formulas tell us how to compute the graded R-modules H∗(X ×Y ; R)
or H∗
(X × Y ; R) out of the graded modules H∗(X; R) and H∗(Y ; R) or H∗
(X; R) and
H∗
(Y ; R), respectively. Under certain conditions it even determines the ring structure
of H∗
(X × Y ; R).
48
Theorem (K¨unneth formula). Let (X, A) and (Y, B) be pairs of CW-complexes. Suppose
that Hk
(Y, B; R) are free finitely generated R-modules for all k. Then
µ : H∗
(X, A; R) ⊗ H∗
(Y, B; R) → H∗
(X × Y, A × Y ∪ X × B; R)
is an isomorphism of graded rings.
Example. H∗
(Sk
× Sl
) ∼= Z[α, β]/I where I is the ideal generated by elements α2
,
β2
, αβ = (−1)kl
βα and deg α = k, deg β = l.
Proof. Consider the diagram
H∗
(X, A) ⊗R H∗
(Y ) //
µ

H∗
(X) ⊗R H∗
(Y )
uujjjjjjjjjjjjjjj
µ

H∗
(A) ⊗R H∗
(Y )
δ∗⊗id
kkVVVVVVVVVVVVVVVVVV
µ

H∗
(X × Y, A × Y )H∗
(X × Y ) // H∗
(X × Y )
uujjjjjjjjjjjjjjj
H∗
(A × Y )
δ∗
kkVVVVVVVVVVVVVVVVVVV
where the upper and the lower triangles come from the long exact sequences for pairs
(X, A) and (X ×Y, A×Y ), respectively. The right rhomb commutes as a consequence
of the naturality of the cross product. We prove that the left rhomb also commutes.
Let ϕ and ψ be cocycles in C∗
(A) and C∗
(Y ), respectively. Let Φ be a cocycle in
C∗
(X) extending ϕ. Then p∗
1Φ ∪ p∗
2ψ ∈ C∗
(X × Y ) extends p∗
1ϕ ∪ p∗
2ψ ∈ C∗
(A × Y ).
Using the definition of the connecting homomorphism in cohomology (see Remark 5.6
B) we get
µ (δ∗
⊗ id)([ϕ] ⊗ [ψ]) = µ[δΦ ⊗ ψ] = p∗
1[δΦ] ∪ p∗
2[ψ],
δ∗
µ([ϕ] ⊗ [ψ]) = δ∗
[p∗
1ϕ ∪ p∗
2ψ] = [δ(p∗
1Φ ∪ p∗
2ψ)] = p∗
1[δΦ] ∪ p∗
2[ψ].
First, we prove the statement of Theorem for a finetedimensional CW-complex X
and A = B = ∅ using the induction by the dimension of X and Five Lemma. If
dim X = 0, X is a finite discrete set and the statement of Theorem is true. Suppose
that Theorem holds for spaces of dimension n − 1 or less. Let dim X = n. It suffices
to show that
µ : H∗
(Xn
, Xn−1
) ⊗ H∗
(Y ) → H∗
(Xn
× Y, Xn−1
× Y )
is an isomorphism and than to use Five Lemma in the diagram above with A = Xn−1
to prove the statement for X = Xn
. Xn
/Xn−1
is homeomorphic to i Dn
i / i ∂Dn
i .
To prove that
µ : H∗
i
Sn
i ⊗ H∗
(Y ) → H∗
i
Sn
i × Y
is an isomorphism, we use again the diagram above for X = i Dn
i and A = i ∂Dn
i
and the induction with respect to n.
49
So we have proved the theorem for X a finite dimensional CW-complex and A =
B = ∅. Using once more our diagram and Five Lemma, we can easily prove Theorem
for any pairs (X, A), (Y, ∅) with X of finite dimension. For X of infinite dimension,
we have to prove Hi
(X) = Hi
(Xn
) for i < n which is equivalent to Hi
(X/Xn
) = 0.
We omit the details and refer the reader to [Hatcher], pages 220 – 221.
7.6. Application of the cup product. In this paragraph we show how to use the
cup product to prove that S2k
is not an H-space. A space X is called an H-space if
there is a map m : X × X → X called a multiplication and an element e ∈ X called
a unit such that m(e, x) = m(x, e) = x for all x ∈ X.
Suppose that there is a multiplication m : S2k
×S2k
→ S2k
with a unit e. According
to Example after Theorem 7.5
H∗
(S2k
× S2k
; Z) = Z[α, β]/I
where I is the ideal generated by relations α2
= 0, β2
= 0 and αβ = βα. The
last relation is due to the fact that the dimension of the sphere is even. Moreover,
α = γ ⊗ 1 and β = 1 ⊗ γ where γ ∈ H2k
(S2k
; Z) is a generator. Let us compute
m∗
: H∗
(S2k
; Z) → H∗
(S2k
× S2k
; Z). We have
m∗
(γ) = aα + bβ, a, b ∈ Z.
Since the composition
S2k id ×e
−−−→ S2k
× S2k m
−−→ S2k
is the identity, we get that a = 1. Similarly, b = 1. Now compute m∗
(γ2
):
0 = m∗
(0) = m∗
(γ2
) = m∗
(γ)
2
= (α + β)2
= 2αβ = 0,
a contradiction. Does this proof go through for odd dimensional spheres?
7.7. K¨unneth formula in homological algebra. Consider two chain complexes
(C∗, ∂C), (D∗, ∂D) of R-modules. Suppose there is an integer N such that Cn = Dn = 0
for all n < N. Then their tensor product is the chain complex (C∗ ⊗ D∗, ∂) with
(C∗ ⊗ D∗)n =
i+j=n
Ci ⊗ Di
and the boundary operator on Ci ⊗ Dj
∂(c ⊗ d) = ∂Cc ⊗ d + (−1)i
c ⊗ ∂Dd.
It is easy to make sure that ∂∂ = 0.
Next we can define the graded R-module C∗ ∗ D∗ as
(C∗ ∗ D∗)n =
i+j=n
TorR
1 (Ci, Dj).
A ring R is called hereditary if any submodule of a free R-module is free. Examples
of hereditary rings are Z and all fields.
50
Theorem (Algebraic K¨unneth formula). Let R be a hereditary ring and let C∗ and D∗
be chain complexes of R-modules. If C∗ is free, then the homology groups of C∗ ⊗ D∗
are determined by the splitting short exact sequence
0 → (H∗(C) ⊗ H∗(D))n
l
−→ Hn(C∗ ⊗ D∗) → (H∗(C) ∗ H∗(D))n−1 → 0
where l([c] ⊗ [d]) = [c ⊗ d]. This sequence is natural but the splitting is not.
Notice that for the chain complex
Dn =
0 for n = 0,
G for n = 0
the K¨uneth formula gives the universal coefficient theorem for homology groups, see
Theorem 6.11 B.
The proof of the K¨unneth formula is similar to the proof of the universal coefficient
theorem and we omit it.
7.8. Eilenberg-Zilbert Theorem. To be able to apply the previous K¨unneth formula
in topology we have to show that the singular chain complex C∗(X × Y ) of a
product X × Y is chain homotopy equivalent to the tensor product of the singular
chain complexes C∗(X) ⊗ C∗(Y ).
Theorem (Eilenberg-Zilbert). Up to chain homotopy there are unique natural chain
homomorphisms
Φ :C∗(X) ⊗ C∗(Y ) → C∗(X × Y ),
Ψ :C∗(X × Y ) → C∗(X) ⊗ C∗(Y )
such that for 0-simplices σ and τ
Φ(σ ⊗ τ) = (σ, τ), Ψ(σ, τ) = σ ⊗ τ.
Moreover, such chain homomorphisms are chain homotopy equivances: there are
natural chain homotopies such that
ΨΦ ∼ idC∗X⊗C∗(Y ), ΦΨ ∼ idC∗(X×Y ) .
For the proof of this theorem see [Dold], IV.12.1. The chain homomorphism Ψ is
called the Eilenberg-Zilbert homomorphism and denoted EZ. It enables a different
and more abstract approach to the definitions of the cross and cup products. The
cross product is
µ([α] ⊗ [β]) = [(α ⊗ β) ◦ EZ]
for cocycles α ∈ C∗
(X; R) and β ∈ C∗
(Y ; R) and the cup product is
([ϕ] ⊗ [ψ]) = [(ϕ ⊗ ψ) ◦ EZ ◦ ∆∗]
for cocycles ϕ, ψ ∈ C∗
(X; R) and the diagonal ∆ : X → X × X. In our definition in
7.1 we have used for EZ ◦ ∆∗ the homomorphism
σ → σ/[v0, v1, . . . , vk] ⊗ σ/[vk, . . . , vn].
The properties of EZ can be used for a different proof of the graded commutativity
of the cup product.
51
7.9. K¨unneth formulas in topology. The following statement is an immediate
consequence of the previous paragraph.
Theorem A (K¨unneth formula for homology). Let R be a hereditary ring. The homology
groups of the product of two spaces X and Y are determined by the following
splitting short exact sequence
0 → (H∗(X; R) ⊗ H∗(Y ; R))n
l
−→ Hn(X × Y ; R) → (H∗(X; R) ∗ H∗(Y ; R))n−1 → 0
where l([c] ⊗ [d]) = [c ⊗ d]. This sequence is natural but the splitting is not.
For cohomology groups one can prove
Theorem B (K¨unneth formula for cohomology groups). Let R be a hereditary ring.
The cohomology groups of the product of two spaces X and Y are determined by the
following splitting short exact sequence
0 → (H∗
(X; R) ⊗ H∗
(Y ; R))n
µ
−→ Hn
(X × Y ; R) → (H∗
(X; R) ∗ H∗
(Y ; R))n+1 → 0.
This sequence is natural but the splitting is not.
For the proof and other forms of K¨unneth formulas see [Dold], Chapter VI, Theorem
12.16 or [Spanier], Chapter 5, Theorems 5.11. and 5.12.
52
8. Vector bundles and Thom isomorphism
In this section we introduce the notion of vector bundle and define its important
algebraic invariants Thom and Euler classes. The Thom class is involved in so called
Thom isomorphism. Using this isomorphism we derive the Gysin exact sequence which
is an important tool for computing cup product structure in cohomology.
8.1. Fibre bundles. A fibre bundle structure on a space E, with fiber F, consists
of a projection map p : E → B such that each point of B has a neighbourhood U for
which there is a homeomorphism h : p−1
(U) → U × F such that the diagram
p−1
(U)
h //
p
##FFFFFFFFF
U × F
pr1
||xxxxxxxxx
U
commutes. Here pr1 is the projection on the first factor. h is called a local trivialization,
the space E is called the total space of the bundle and B is the base space.
A subbundle (E , B, p ) of a fibre bundle (E, B, p) has the total space E ⊆ E,
the fibre F ⊆ F, p = p/E and local trivializations in E are restrictions of local
trivializations of E.
A vector bundle is a fibre bundle (E, B, p) whose fiber is a vector space (real or
complex). Moreover, we suppose that for each b ∈ B the fiber p−1
(b) over b is a vector
space and all local trivializations restricted to p−1
(b) are linear isomorphisms. The
dimension of a vector bundle is the dimension of its fiber. For p−1
(U) where U ⊆ B
we will use notation EU . Further, E0
U will stand for EU without zeroes in vector spaces
Ex = p−1
(x) for x ∈ U.
8.2. Orientation of a vector space. Let V be a real vector space of dimension n.
The orientation of V is the choice of a generator in Hn
(V, V − {0}; Z) = Z. If R is
a commutative ring with a unit, the R-orientation of V is the choice of a generator
in Hn
(V, V − {0}; R) = R. For R = Z we have two possible orientations, for R = Z2
only one.
8.3. Orientation of a vector bundle. Consider a vector bundle (E, B, p) with fiber
Rn
. The R-orientation of the vector bundle E is a choice of orientation in each vector
space p−1
(b), b ∈ B, i. e. a choice of generators tb ∈ Hn
(Eb, E0
b ; R) = R such that for
each b ∈ B there is a neighbourhood U and an element
tU ∈ Hn
(EU , E0
U ; R)
with the property
i∗
x(tU ) = tx
for each x ∈ U and the inclusion ix : Ex → EU .
53
If a vector bundle has an R-orientation, we say that it is R-orientable. An R-oriented
vector bundle is a vector bundle with a chosen R-orientation. Talking on orientation
we will mean Z-orientation.
Example. Every vector bundle (E, B, p) is Z2-orientable. After we have some knowledge
of fundamental group, we will be able to prove that vector bundles with π1(B) = 0
are orientable.
8.4. Thom class and Thom isomorphism. The Thom class of a vector bundle
(E, B, p) of dimension n is an element t ∈ Hn
(E, E0
; R) such that i∗
b(t) is a generator
in Hn
(Eb, E0
b ; R) = R for each b ∈ B where ib : Eb → E is an inclusion.
It is clear that any Thom class determines a unique orientation. The reverse statement
is also true.
Theorem (Thom Isomorphism Theorem). Let (E, B, p) be an R-oriented vector bundle
of real dimension n. Then there is just one Thom class t ∈ Hn
(E, E0
; R) which
determines the given R-orientation. Moreover, the homomorphism
τ : Hk
(B; R) → Hk+n
(E, E0
; R), τ(a) = p∗
(a) ∪ t
is an isomorphism (so called Thom isomorphism).
Remark. Notice that Thom Isomorphism Theorem is a generalization of the K¨unneth
Formula 7.5 for (Y, A) = (Rn
, Rn
− {0}). We use it in the proof.
Proof. (1) First suppose that E = B × Rn
. Then according to Theorem 7.5
H∗
(E, E0
; R) = H∗
(B × Rn
, B × (Rn
− {0}); R) = H∗
(B; R) ⊗ H∗
(Rn
, Rn
− {0}); R)
∼= H∗
(B; R)[e]/ e2
where e ∈ Hn
(Rn
, Rn
− {0}); R) is the generator given by the orientation of E. Now,
there is just one Thom class t = 1 × e and
τ(a) = p∗
(a) ∪ t = a × e
is an isomorphism.
(2) If U is open subset of B, then the orientation of (E, B, p) induces an orientation of
the vector bundle (EU , U, p). Suppose that U and V are two open subsets in B such
that the statement of Theorem is true for EU , EV and EU∩V with induced orientations.
Denote the corresponding Thom classes by tU , tV and tU∩V . The uniqueness of tU∩V
implies that the restrictions of both classes tU and tV on Hn
(EU∩V , E0
U∩V ; R) are tU∩V .
We will show that Theorem holds for EU∪V .
Consider the Mayer-Vietoris exact sequence 5.13 for A = EU , B = EV , C = E0
U ,
D = E0
V together with the Mayer-Vietoris exact sequence for A = U, B = V and C =
D = ∅. Omitting coefficients these exact sequences together with Thom isomorphisms
54
τU , τV and τU∩V yield the following diagram where DEU stands for the pair (EU , E0
U )
δ∗
// Hk+n
(DEU∪V )
(j∗
U ,j∗
V )
// Hk+n
(DEU ) ⊕ Hk+n
(DEV )
i∗
U −i∗
V
// Hk+n
(DEU∩V ) //
δ∗
// Hk
(U ∪ V )
(j∗
U ,j∗
V )
//
τU∪V
OO


Hk
(U) ⊕ Hk
(V )
i∗
U −i∗
V
//
τU ⊕τV
OO
Hk
(U ∩ V )
τU∩V
OO
//
(At the moment we do not need commutativity.) From the first row of this diagram
we get that
Hi
(EU∪V , E0
U∪V ) = 0 for i < n
and that there is just one class tU∪V ∈ Hn
(EU∪V , E0
U∪V ) such that
(j∗
U , j∗
V )(tU∪V ) = (tU , tV ).
This is the Thom class for EU∪V and we can define the homomorphism τU∪V : Hk
(U ∪
V ) → Hk+n
(EU∪V , E0
U∪V ) by
τU∪V (a) = p∗(a) ∪ tU∪V .
Complete the diagram by this homomorphism. When we check the commutativity
of the completed diagram, it suffices to apply Five Lemma to show that τU∪V is an
isomorphism.
To prove the commutativity we have to go into the cochain level from which the
Mayer-Vietoris sequences are derived in natural way. Let tU and tV be cocycles representing
the Thom classes tU and tV . We can choose them in such a way that
i∗
U tU = i∗
V tV = tU∩V
where tU∩V represents the Thom class tU∩V . Consider the diagram where the rows are
the short exact sequences inducing the Mayer-Vietoris exact sequences above.
0 // C∗
0 (EU + EV )
(j∗
U ,j∗
V )
// C∗
0 (EU ) ⊕ C∗
0 (EV )
i∗
U −i∗
V
// C∗
(EU∩V ) // 0
0 // C∗
(U + V )
(j∗
U ,j∗
V )
//
τU∪V
OO


C∗
(U) ⊕ C∗
(V )
τU ⊕τV
OO
i∗
U −i∗
V
// C∗
(U ∩ V ) //
τU∩V
OO
0
Here we use the following notation: C∗(U + V ) is the free Abelian group generated
by simplices lying in U and V , C∗
(U + V ) = HomR(C∗(U + V ), R). C∗
0 (EU + EV )
are the cochains from C∗
(EU + EV ) which are zeroes on simplices from C∗(E0
U + E0
V ).
τU (a) = p∗
(a) ∪ tU . (According to Lemma in 3.12 the cohomology of C∗
0 (EU + EV ) is
H∗
(EU∪V , E0
U∪V ; R).)
There is just one cocycle tU∪V representing the Thom class tU∪V such that
(j∗
U , j∗
V )(tU∪V ) = (tU , tV ).
If we show that τU , τV , τU∩V and τU∪V are cochain homomorphisms which make the
diagram commutative, then the diagram with the Mayer-Vietoris exact sequences will
be also commutative. To prove the commutativity of the cochain diagram above is
not difficult and left to the reader. Here we prove that τU is a cochain homomorhism.
(The proof for the other τ is the same.)
55
Let a ∈ Ck
(U). Since tU is cocycle we get
δτU (a) = δ(p∗
(a) ∪ tU ) = δ(p∗
(a)) ∪ tU + (−1)k
p∗
(a) ∪ δ(tU ) = p∗
(δ(a)) ∪ tU = τU δ(a).
(3) Let B be compact (particullary a finite CW-complex). Then there is a finite open
covering U1, U2, . . . , Um such that EUi
is homeomorphic to Ui × Rn
. So according to
(1) the statement of Theorem holds for all EUi
. Using (2) and induction we can show
that Theorem holds for E = m
i=1 EUi
as well.
(4) The proof for the other base spaces B needs a limit transitions in cohomology and
the fact that for any B there is always a CW-complex X and a map f : B → X
inducing isomorphism in cohomology. Here we omit this part.
8.5. Euler class. Let ξ = (E, B, p) be oriented vector bundle of dimension n with
the Thom class tξ ∈ Hn
(E, E0
; Z). Consider the standard inclusion j : E → (E, E0
).
Since p : E → B is a homotopy equivalence, there is just one class e(ξ) ∈ Hn
(B; Z),
called the Euler class of ξ, such that
p∗
(e(ξ)) = j∗
(tξ).
For R-oriented vector bundles we can define the Euler class e(ξ) ∈ Hn
(B; R) in
the same way. Particulary, any vector bundle ξ = (E, B, p) has an Euler class with
Z2-coefficients called the n-th Stiefel-Whitney class wn(ξ) ∈ Hn
(B; Z2).
8.6. Gysin exact sequence. The following theorem gives us a useful tool for computation
of the ring structure of singular cohomology of various spaces.
Theorem (Gysin exact sequence). Let ξ = (E, B, p) be an R-oriented vector bundle
of dimension n with the Euler class e(ξ) ∈ Hn
(B; R). Then there is a homomorphism
∆∗
: H∗
(E0
; R) → H∗
(B; R) of modules over H∗
(B; R) such that the sequence
. . .
p∗
−→ Hk+n−1
(E0
; R)
∆∗
−→ Hk
(B; R)
∪e(ξ)
−−−→ Hk+n
(B; R)
p∗
−→ Hk+n
(E0
; R)
∆∗
−→ . . .
is exact.
Proof. The definition of ∆∗
and the exactness follows from the following cummutative
diagram where we have used the long exact sequence for the pair (E, E0
) and the
Thom isomorphism τ:
Hk+n−1
(E0
)
δ∗
//
∆∗
((PPPPPP
Hk+n
(E, E0
)
j∗
// Hk+n
(E)
i∗
// Hk+n
(E0
)
Hk
(B)
τ∼=
OO
∪e(ξ)
//____ Hk+n
(B)
p∗∼=
OO
p∗
88qqqqqqqqqq
The right action of b ∈ H∗
(B) on H∗
(E0
) is given by
x · b = x ∪ i∗
p∗
(b), x ∈ H∗
(E0
).
Using the definition of the connecting homomorphism and the properties of cup product
one can show that
∆∗
(x · b) = ∆∗
(x) ∪ b.
56
The details are left to the reader.
Example. Consider the canonical one dimensional vector bundle γ = (E, RPn
, p)
where
E = {(l, v) ∈ RPn
× Rn+1
; v ∈ l},
the elements of RPn
are identified with lines in Rn+1
and p(l, v) = l. The space E0 is
equal to Rn+1
− {0} and homotopy equivalent to Sn
.
Using the Gysin exact sequence with Z2-coefficients and the fact that Hk
(RPn
; Z2) =
Z2 for 0 ≤ k ≤ n, we get successively that the first Stiefel-Whitney class w1(γ) ∈
H1
(RPn
; Z2) is different from zero and that
H∗
(RPn
); Z2) ∼= Z2[w1(γ)]/ w1(γ)n+1
.
Exercise. Using the Gysin exact sequence show that
H∗
(CPn
; Z) ∼= Z[x]/ xn+1
where x ∈ H2
(CPn
; Z).
57
9. Poincar´e duality
Many interesting spaces used in geometry are closed oriented manifolds. Poincar´e
duality expresses a remarkable symmetry between their homology and cohomology.
9.1. Manifolds. A manifold of dimension n is a Hausdorff space M in which each
point has an open neighbourhood U homeomorphic to Rn
. The dimension of M is
characterized by the fact that for each x ∈ M, the local homology group Hi(M, M −
{x}; Z) is nonzero only for i = n since by excision and homotopy equivalence
Hi(M, M − {x}; Z) ∼= Hi(U, U − {x}; Z) ∼= Hi(Rn
, Rn
− {0}; Z) ∼= ˜Hi−1(Sn−1
; Z).
A compact manifold is called closed.
Example. Examples of closed manifolds are spheres, real and complex projective
spaces, orthogonal groups O(n) and SO(n), unitary groups U(n) and SU(n), real and
complex Stiefel and Grassmann manifolds. The real Stiefel manifold Vn,k is the space
of k-tuples of orthonormal vectors in Rn
. The real Grassmann manifolds Gn,k is the
space of k-dimensional vector subspaces of Rn
.
9.2. Orientation of manifolds. Consider a manifold M of dimension n. A local
orientation of M in a point x ∈ M is a choice of a generator µx ∈ Hn(M, M−{x}; Z) ∼=
Z.
To shorten our notation we will use Hi(M|A) for Hi(M, M − A; Z) and Hi
(M|A)
for Hi
(M; M − A; Z) if A ⊆ M.
An orientation of M is a function assigning to each point x ∈ M a local orientation
µx ∈ Hn(M|x) such that each point has an open neighbourhood B with the property
that all local orientations µy for y ∈ B are images of an element µB ∈ Hn(M|B) under
the map ρy∗ : Hn(M|B) → Hn(M|x) where ρy : (M, M − {x}) → (M, M − B) is the
natural inclusion.
If an orientation exists on M, the manifold is called orientable. A manifold with a
chosen orientation is called oriented.
Proposition. A connected manifold M is orientable if it is simply connected, i. e.
every map S1
→ M is homotopic to a constant map.
For the proof one has to know more about covering spaces and fundamental group.
See [Hatcher], Proposition 3.25, pages 234 – 235.
In the same way we can define an R-orientation of a manifold for any commutative
ring R. Every manifold is Z2-oriented.
9.3. Fundamental class. A fundamental class of a manifold M with coefficients in
R is an element µ ∈ Hn(M; R) such that ρx∗(µ) is a generator of Hn(M|x; R) = R for
each x ∈ M where ρx : (M, ∅) → (M, M − {x}) is the obvious inclusion. It is usual to
denote the fundamental class of the manifold M by [M]. We will keep this notation.
If a fundamental class of M exists, it determines uniquely the orientation µx =
ρx∗([M]) of M.
58
Theorem. Let M be a closed manifold of dimension n. Then:
(a) If M is R-orientable, the natural map Hn(M; R) → Hn(M|x; R) = R is an
isomorphism for all x ∈ M.
(b) If M is not R-orientable, the natural map Hn(M; R) → Hn(M|x; R) = R is
injective with the image {r ∈ R; 2r = 0} for all x ∈ M.
(c) Hi(M; R) = 0 for all i > n.
(a) implies immediately that very oriented closed manifold has just one fundamental
class. It is a suitable generator of Hn(M; R).
The theorem will follow from a more technical statement:
Lemma. Let M be n-manifold and let A ⊆ M be compact. Then:
(a) Hi(M|A; R) = 0 for i > n and α ∈ Hn(M|A; R) is zero iff its image ρx∗(α) ∈
Hn(M|x; R) is zero for all x ∈ M.
(b) If x → µx is an R-orientation of M, then there is µA ∈ Hn(M|A; R) whose
image in Hn(M|x; R) is µx for all x ∈ A.
To prove the theorem put A = M. We get immediately (c) of the theorem. Further,
the lemma implies that an oriented manifold M has a fundamental class [M] = µM
and any other element in Hn(M; R) has to be its multiple in R. So we obtain (a) of
the theorem. For the proof of (b) we refer to [Hatcher], pages 234 – 236.
Proof of Lemma. Since R does not play any substantial role in our considerations, we
will omit it from our notation. We will omit also stars in notation of maps induced in
homology. The proof will be divided into several steps.
(1) Suppose that the statements are true for compact subsets A, B and A ∩ B of M.
We will prove them for A ∪ B using the Mayer-Vietoris exact sequence:
0 → Hn(M|A ∪ B)
Φ
−→ Hn(M|A) ⊕ Hn(M|B)
Ψ
−→ Hn(M|A ∩ B)
where Φ(α) = (ρAα, ρBα), Ψ(α, β) = ρA∩Bα − ρA∩Bβ.
Hi(M|A ∪ B) = 0 for i > n is immediate from the exact sequence. Suppose α ∈
Hn(M|A ∪ B) restricted to Hn(M|x) is zero for all x ∈ A ∪ B. Then ρAα and ρBα are
zeroes. Since Φ is a monomorphism, α has to be also zero.
Take µA and µB such that their restrictions to Hn(M|x) are orientations. Then the
restrictions to points x ∈ A ∩ B are the same. Hence also the restrictions to A ∩ B
coincide. It means Ψ(µA, µB) = 0 and the Mayer-Vietoris exact sequence yields the
existence of α in Hn(M|A ∪ B) such that Φ(α) = (µA, µB). Therefore α reduces to a
generator of Hn(M|x) for all x ∈ A ∪ B, and consequently, α = µA∪B.
(2) If M = Rn
and A is a compact convex set in a disc D containing an origin 0, the
lemma is true since the composition given by inclusions
Hi(Rn
|D) −→ Hi(Rn
|A) −→ Hi(Rn
|0)
is an isomorhism.
(3) If M = Rn
and A is finite simplicial complex in Rn
, then A = m
i=1 Ai where Ai
are convex compact sets. Using (1) and induction by m we can prove that the lemma
holds in this case as well.
59
(4) Let M = Rn
and A is an arbitrary compact subset. Let α ∈ Hi(Rn
|A) be represented
by a relative cycle z ∈ Zi(Rn
, Rn
− A). Let C ⊂ Rn
− A be the union of images
of the singular simplices in ∂z. Since C is compact, dist(C, A) > 0, and consequenly,
there is a finite simplicial complex K ⊃ A such that C ⊂ Rn
− K. (Draw a pisture.)
So the chain z defines also an element αK ∈ Hi(Rn
|K) which reduces to α ∈ Hi(Rn
|A).
If i > n, then by (3) αK = 0 and consequently also α = 0.
Suppose that i = n and that α reduces to zero in each point x ∈ A. K can be
chosen in such a way that every its point lies in a simplex of K together with a point
of A. Consequently, αK reduces to zero not only for all x ∈ A but for all x ∈ K. (Use
the case (2) to prove it.) By (3) αK = 0, and therefore also α = 0.
The proof of existence of µA ∈ Hn(Rn
|A) in the statement (b) is easy. Take µB ∈
Hn(Rn
|B) for a ball B ⊃ A and its reduction is µA.
(5) Let M be a general manifold and A a compact subset in an open set U homeomorphic
to Rn
. Now by excision
Hi(M|A) ∼= Hi(U|A) ∼= Hi(Rn
|A)
and we can use (4).
(6) Let M be a manifold and A an arbitrary compact set. Then A can be covered by
open sets V1, V2, . . . , Vm such that the closure of Vi lies in an open set Ui homeomorphic
to Rn
. Then by (5) the lemma holds for Ai = A ∩ ¯Vi. By (1) and induction it holds
also for m
i=1 Ai = A.
9.4. Cap product. Let X be a space. On the level of chains and cochains the cap
product
∩ : Cn(X; R) ⊗ Ck
(X; R) → Cn−k(X; R)
is given for 0 ≤ k ≤ n by
σ ∩ ϕ = ϕ(σ/[v0, v1, . . . , vk])σ/[vk, vk+1, . . . , vn]
where σ is a singular n-simplex, ϕ : Ck(X; R) → R is a cochain and σ/[v0, v1, . . . , vk]
is the composition of the inclusion of ∆k
into the indicated face of ∆n
with σ, and is
given by zero in the remaining cases.
The proof of the following statement is similar as in the case of cup product and is
left to the reader as an exercise.
Lemma A. For σ ∈ Cn(X; R) and ϕ ∈ Ck
(X; R)
∂(σ ∩ ϕ) = (−1)k
(∂σ ∩ ϕ − σ ∩ δϕ)
This enables us to define
∩ : Hn(X; R) ⊗ Hk
(X; R) → Hn−k(X; R)
by
[σ] ∩ [ϕ] = [σ ∩ ϕ]
60
for all cycles σ and cocycles ϕ. In the same way one can define
∩ : Hn(X, A; R) ⊗ Hk
(X; R) → Hn−k(X, A; R)
∩ : Hn(X, A; R) ⊗ Hk
(X, A; R) → Hn−k(X; R)
for any pair (X, A) and
∩ : Hn(X, A ∪ B; R) ⊗ Hk
(X, A; R) → Hn−k(X, B; R)
for A, B open in X or subcomplexes of CW-complex X.
Exercise. Show the correctness of all the definitions above and prove the following
lemma.
Lemma B (Naturality of cup product). Let f : (X, A) → (Y, B). Then
f∗(α ∩ f∗
(β)) = f∗(α) ∩ β
for all α ∈ Hn(X, A; R) and β ∈ Hk
(Y ; R).
9.5. Poincar´e duality. Now we have all the tools needed to state the Poincar´e
duality for closed manifolds.
Theorem (Poincar´e duality). If M is a closed R-orientable manifold of dimension n
with fundamental class [M] ∈ Hn(M; R), then the map D : Hk
(M; R) → Hn−k(M; R)
defined by
D(ϕ) = [M] ∩ ϕ
is an isomorphism.
Exercise. Use Poincar´e duality to show that the real projective spaces of even dimension
are not orientable.
This theorem is a consequence of a more general version of Poincar´e duality. To
state it we introduce the notion of direct limit and cohomology with compact support.
9.6. Direct limits. A direct set is a partially ordered set I such that for each pair
ι, κ ∈ I there is λ ∈ I such that ι ≤ λ and κ ≤ λ.
Let Gι be a system od Abelian groups (or R-modules) indexed by elements of a
directed set I. Suppose that for each pair ι ≤ κ of indices there is a homomorphism
fικ : Gι → Gκ such that fιι = id and fκλfικ = fιλ. Then such a system is called
directed.
Having a directed system of Abelian groups (or R-modules) we will say that a ∈ Gι
and b ∈ Gκ are equivalent (a b) if fιλ(a) = fκλ(b) for some λ ∈ I. The direct limit
of the system {Gι}ι∈I is the Abelian group (R-module) of classes of this equivalence
lim−→Gι =
ι∈I
Gι/ .
Moreover, we have natural homomorphism jι : Gι → lim−→Gι.
61
The direct limit is characterized by the following universal property: Having a
system of homomorphism hι : Gι → A such that hι = hκfικ whenever ι ≤ κ, there is
just one homomorphism
H : lim−→Gι → A
such that hι = Hjι.
It is not difficult to prove that direct limits preserve exact sequences.
In a system of sets the ordering is usually given by inclusions.
Lemma. If a space X is the union of a directed set of subspaces Xι with the property
that each compact set in X is contained in some Xι, the natural map
lim−→ Hn(Xι; R) → Hn(X; R)
is an isomorphism.
The proof is not difficult, we refer to [Hatcher], Proposition 3.33, page 244.
9.7. Cohomology groups with compact support. Consider a space X with a
directed system of compact subsets. For each pair (L, K), K ⊆ L, the inclusion
(X, X − L) → (X, X − K) induces homomorphism Hk
(X|K; R) → Hk
(X|L; R). We
define the cohomology groups with compact support as
Hk
c (X; R) = lim−→ Hk
(X|K; R).
If X is compact, then Hk
c (X; R) = Hk
(X; R).
For cohomology with compact support we get the following lemma which does not
hold for ordinary cohomology groups.
Lemma. If a space X is the union of a directed set of open subspaces Xι with the
property that each compact set in X is contained in some Xι, the natural map
lim−→ Hk
c (Xι; R) → Hk
c (X; R)
is an isomorphism.
Proof. The definition of natural homomorphism in the lemma is based on the following
fact: Let U be an open subset in V . For any compact set K ⊂ U the inclusion
(U, U − K) → (V, V − K) induces by excision an isomorphism
Hk
(V |K; R) → Hk
(U|K; R).
Its inverse can be composed with natural homomorphism Hk
(V |K; R) → Hk
c (V ; R).
By the universal property of direct sum there is just one homomorphism
Hk
c (U; R) → Hk
c (V ; R).
So on inclusions of open sets Hk
c behaves as covariant functor and this makes the
definition of the natural homomorphism in the lemma possible. The proof that it is
an isomorphism (based on excision) is left to the reader.
62
9.8. Generalized Poincar´e duality. Let M be an R-orientable manifold of dimension
n. Let K ⊆ M be compact. Let µK ∈ Hn(M|K; R) be such a class that its
reduction to Hn(M|x; R) gives a generator for each x ∈ K. The existence of such a
class is ensured by Lemma in 9.3. Define
DK : Hk
(M|K) → Hn−k(M; R) : DK(ϕ) = µK ∩ ϕ.
If K ⊂ L are two compact subsets of M, we can easily prove using naturality of cap
product that
DL(ρ∗
ϕ) = DK(ϕ)
for ϕ ∈ Hk
(M|K; R) and ρ : (M, M − L) → (M, M − K). It enables us to define the
generalized duality map
DM : Hk
c (M; R) → Hn−k(M; R) : DM (ϕ) = µK ∩ ϕ
since each element ϕ ∈ Hk
c (M; R) is contained in Hk
(M|K; R) for some compact set
K ⊆ M.
Theorem (Duality for all orientable manifolds). If M is an R-orientable manifold of
dimension n, then the duality map
DM : Hk
c (M; R) → Hn−k(M; R)
is an isomorphism.
The proof is based on the following
Lemma. If a manifolds M be a union of two open subsets U and V , the following
diagram of Mayer-Vietoris sequences
Hk
c (U ∩ V ) //
DU∩V

Hk
c (U) ⊕ Hk
c (V ) //
DU ⊕DV

Hk
c (M) //
DM

Hk+1
c (U ∩ V )
DU∩V

Hn−k(U ∩ V ) // Hn−k(U) ⊕ Hn−k(V ) // Hn−k(M) // Hn−k−1(U ∩ V )
commutes up to signs.
The proof of this lemma is analogous as the proof of commutativity of the diagram
in the proof of Theorem 8.4 on Thom isomorphism. So we omit it referring the reader
to [Hatcher], Lemma 3.36, pages 246 – 247 or to [Bredon], Chapter VI, Lemma 8.2,
pages 350 – 351.
Proof of Poincar´e Duality Theorem. We will use the following two statements
(A) If M = U ∪ V where U and V are open subsets such that DU , DV and DU∩V
are isomorphisms, then DM is also an isomorphism.
(B) If M = ∞
i=1 Ui where Ui are open subsets such that U1 ⊂ U2 ⊂ U3 ⊂ . . . and
all DUi
are isomorphisms, then DM is also an isomorphism.
The former is an immediate consequence of the previous lemma and Five Lemma. To
obtain the latter apply the direct limit to the short exact sequences
0 → Hk
c (Ui)
DUi
−−→ Hn−k(Ui) → 0
63
and use the lemmas in 9.6 and 9.7. The proof of Duality Theorem will be carried out
in four steps.
(1) For M = Rn
we have
Hk
c (Rn
) ∼= Hk
(∆n
, ∂∆n
), Hn(Rn
|∆n
) ∼= Hn(∆n
, ∂∆n
).
Take the generator µ ∈ Hn(∆n
, ∂∆n
) represented by the singular simplex given by
identity. The only nontriavial case is k = n. In this case for a generator
ϕ ∈ Hn
((∆n
∂∆n
)) = Hom(Hn((∆n
∂∆n
), R)
we get µ ∩ ϕ = ϕ(µ) = ±1. So the duality map is an isomorphism.
(2) Let M ⊂ Rn
be open. Then M is a countable union of open convex sets Vi which
are homeomorphic to Rn
. Using the previous step and induction in statement (A)
we show that the duality map is an isomorphism for every finite union of Vi. The
application of statement (B) yields that the duality map DM is an isomorphism as
well.
(3) Let M be a manifold which is a countable union of open sets Ui which are homeomorphic
to Rn
. Now we can proceed in the same way as in (2) using its result instead
of the result in (1).
(4) For general M we have to use Zorn lemma. See [Hatcher], page 248.
Corollary. The Euler characteristic of a closed manifold of odd dimension is zero.
Proof. For M orientable we get from Poincar´e duality and the universal coefficient
theorem that
rank Hn−k(M; Z) = rank Hk
(M; Z) = rank Hom Hk(M; Z)
= rank Hk(M; Z)
Hence χ(M) = n
i=0(−1)i
rank Hi(M; Z) = 0 for n odd.
If M is not orientable, we get from the Poincar´e duality with Z2 coefficients that
n
i=0
(−1)i
dim Hi(M; Z2) = 0.
Here the dimension is considered over Z2. Applying the universal coefficient theorem
one can show that the expression on the left hand side equals to χ(M). See [Hatcher],
page 249.
Remark. Consider an oriented closed smooth manifold M. The orientation of the
manifold induces an orientation of the tangent bundle τM and we get the following
relation between the Euler class of τM , the fundamental class of M and the Euler
characteristic of M:
χ(M) = e(τM ) ∩ [M].
Particulary, for spheres of even dimension we get that the Euler class of their tangent
bundle is twice a generator of Hn
(Sn
; Z). For the proof see [MS], Corollary 11.12.
64
9.9. Duality and cup product. One can easily show that for α ∈ Cn(X; R),
ϕ ∈ Ck
(X; R) and ψ ∈ Cn−k
(X; R) we have
ψ(α ∩ ϕ) = (ϕ ∪ ψ)(α).
For a closed R-orientable manifold M we define bilinear form
(∗) Hk
(M; R) × Hn−k
(M; R) → R : (ϕ, ψ) → (ϕ ∪ ψ)[M].
A bilinear form A × B → R is called regular if induced linear maps A → Hom(B, R)
and B → Hom(A, R) are isomorphisms.
Theorem. Let M be a closed R-orientable manifold. If R is a field, then the bilinear
form (∗) is regular.
If R = Z, then the bilinar form
Hk
(M; Z)/ Torsion Hk
(M; Z) × Hn−k
(M; Z)/ Torsion Hn−k
(M; Z) → Z
induced by (∗) is regular.
Proof. Consider the homomorphism
Hn−k
(M; R)
h
−−→ Hom(Hn−k(M; R); R)
D∗
−−→ Hom(Hk
(M; R), R).
Here h(ψ)(β) = ψ(β) for β ∈ Hn−k(M; R) and ψ ∈ Hn−k
(M; R) and D∗
is the dual
map to duality. The homomorphism h is an isomorphism by the universal coefficient
theorem and D∗
is an isomorphism since so is D. Now it suffices to prove that the
composition D∗
h is the homomorphism induced from the bilinear form (∗). For ψ ∈
Hn−k
(M; R) and ϕ ∈ Hk
(M; R) we get
(D∗
h(ψ)) (ϕ) = (h(ψ)) D(ϕ) = (h(ψ)) ([M] ∩ ϕ) = ψ([M] ∩ ϕ) = (ϕ ∪ ψ)[M].
This theorem gives us a further tool for computing the cup product structure in
cohomology of closed manifolds.
Corollary. Let M be a closed orientable manifold of dimension n. Then for every
ϕ ∈ Hk
(M; Z) of infinite order which is not of the form ϕ = mϕ1 for m > 1, there is
ψ ∈ Hn−k
(M; Z) such that ϕ ∪ ψ is a generator of Hn
(M; Z) ∼= Z.
Example. We will prove by induction that H∗
(CPn
; Z) = Z[ω]/ ωn+1
where ω ∈
H2
(CPn
; Z) is a generator. For n = 1 the statement is clear. Suppose that it holds for
n − 1. From the long exact sequence for the pair (CPn
, CPn−1
) we get that
Hi
(CPn
; Z) ∼= Hi
(CPn−1
; Z)
for i ≤ 2n − 1. Now, using the consequence above for ϕ = ω we obtain that ωn
is a
generator of H2n
(CPn
; Z).
9.10. Manifolds with boundary. A manifold with boundary of dimension n is a
Hausdorff space M in which each point has an open neighbourhood homeomorphic
either to Rn
or to the half-space
Rn
+ = {(x1, x2, . . . , xn) ∈ Rn
; xn ≥ 0}.
65
The boundary ∂M of the manifold M is formed by points which have all neighbourhoods
of the second type. The boundary of a manifold of dimension n is a manifold
of dimension n − 1. In a similar way as for a manifold we can define orientation of a
manifold with boundary and its fundamental class [M] ∈ Hn(M; ∂M; R).
Theorem. Suppose that M is a compact R-orientable n-dimensional manifold whose
boundary ∂M is decomposed as a union of two compact (n−1)-dimensional manifolds
A and B with common boundary ∂A = ∂B = A ∩ B. Then the cap product with the
fundamental class [M] ∈ Hn(M, ∂M; R) gives the isomorphism
DM : Hk
(M, A; R) → Hn−k(M, B; R).
For the proof and many other applications of Poincar´e duality we refer to [Hatcher],
Theorem 3.43 and pages 250 – 254, and [Bredon], Chapter VI, Sections 9 and 10, pages
355 – 366.
9.11. Alexander duality. In this paragraph we introduce another version of duality.
Theorem (Alexander duality). If K is a proper compact subset of Sn
which is a
deformation retract of an open neighbourhood, then
˜Hi(Sn
− K; Z) ∼= ˜Hn−i−1
(K; Z).
Proof. For i = 0 and U a neighbourhood of K we have
Hi(Sn
− K) ∼= Hn−i
c (Sn
− K) by Poincar´e duality
∼= lim−→ U Hn−i
(Sn
− K, U − K) by definition
∼= lim−→ U Hn−i
(Sn
, U) by excision
∼= lim−→ U
˜Hn−i−1
(U) connecting homomorphism
∼= ˜Hn−i−1
(K) K is a def. retract of some U
First three isomorphisms are natural and exist also for i = 0. So using these facts
we have
˜H0(Sn
− K) ∼= Ker (H0(Sn
− K) → H0(pt))
∼= Ker (H0(Sn
− K) → H0(Sn
))
∼= Ker lim−→ Hn
(Sn
, U) → Hn
(Sn
)
∼= lim−→ Ker (Hn
(Sn
, U) → Hn
(Sn
))
∼= lim−→ Hn−1
(U) = Hn−1
(K).
Corollary. A closed nonorientable manifold of dimension n cannot be embedded as a
subspace into Rn+1
.
Proof. Suppose that M can be embedded into Rn+1
. Then it can be embedded also
in Sn+1
. By Alexander duality
Hn−1(M; Z) ∼= H1
(Sn+1
− M; Z).
66
According to the universal coefficient theorem
H1
(Sn+1
− M; Z) ∼= Hom(H1(Sn+1
− M; Z), Z) ⊕ Ext(H0(Sn+1
− M; Z))
is a free Abelian group. On the other hand
Z2 = Hn(M; Z2) ∼= Hn(M; Z) ⊗ Z2 ⊕ Tor(Hn−1(M, Z), Z2).
According to (b) of Theorem 9.3 the tensor product has to be zero, and since Hn−1(M; Z)
is free, the second summand has to be also zero, which is a contradiction.
67
10. Homotopy groups
In this section we will define homotopy groups and derive their basic properties.
While the definition of homotopy groups is relatively simple, their computation is
complicated in general.
10.1. Homotopy groups. Let In
be the n-dimensional unit cube and ∂In
its boundary.
For n = 0 we take I0
to be one point and ∂I0
to be empty. Consider a space
X with a basepoint x0. Maps (In
, ∂In
) → (X, x0) are the same as the maps of the
quotient (Sn
= In
/∂In
, s0 = ∂In
/∂In
) → (X, x0). We define the n-th homotopy group
of the space X with the basepoint x0 as
πn(X, x0) = [(Sn
, s0), (X, x0)] = [(In
, ∂In
), (X, x0)].
π0(X, x0) is the set of path connected components of X with the component containing
x0 as a distinguished element. For n ≥ 1 we can introduce a sum operation on πn(X, x0)
(f + g)(t1, t2, . . . , tn) =
f(2t1, t2, . . . , tn) t1 ∈ [0, 1
2
],
g(2t1 − 1, t2, . . . , tn) t1 ∈ [1
2
, 1].
This operation is well defined on homotopy classes. It is easy to show that πn(X, x0)
is a group with identity element represented by the constant map to x0 and with the
inverse represented by
−f(t1, t2, . . . , tn) = f(1 − t1, t2, . . . , tn).
For n ≥ 2 the groups πn(X, x0) are commutative. The proof is indicated by the
following pictures.
f g
f f
f g
g g
Figure 10.1. f + g ∼ g + f
In the interpretation of πn(X, x0) as [(Sn
, s0), (X, x0)], the sum f + g is the compo-
sition
Sn c
−→ Sn
∨ Sn f∨g
−−→ X
where c collapses the equator Sn−1
of Sn
to a point s0 ∈ Sn−1
⊂ Sn
.
Any map F : (X, x0) → (Y, y0) induces the homomorphism F∗ : πn(X, x0) →
πn(Y, y0) by composition
F∗([f]) = [Ff].
Hence πn is a functor from Top∗ to the category of Abelian groups Ab for n ≥ 2, to the
category of groups G for n = 1 and to the category of sets with distiguished element
Set∗ for n = 0.
68
10.2. Relative homotopy groups. Consider In−1
as a face of In
with the last
coordinate tn = 0. Denote Jn−1
the closure of ∂In
− In−1
. Let (X, A) be a pair with
basepoint x0 ∈ A. For n ≥ 1 we define the n-th relative homotopy group of the pair
(X, A) as
πn(X, A, x0) = [(Dn
, Sn−1
, s0), (X, A, x0)] = [(In
, ∂In
, Jn−1
), (X, A, x0)].
A sum operation on πn(X, A, x0) is defined by the same formula as for πn(X, x0) only
for n ≥ 2. (Explain why this definition does not work for n = 1.) Similarly as in the
case of absolute homotopy groups one can show that πn(X, A, x0) is a group for n ≥ 2
which is commutative if n ≥ 3.
Sometimes it is useful to know how the representatives of zero (neutral element)
in πn(X, A, x0) look like. We say that two maps f, g : (Dn
, Sn−1
, s0) → (X, A, x0)
are homotopic rel Sn−1
if there is a homotopy h between f and g such that h(x, t) =
f(x) = g(x) for all x ∈ Sn−1
and all t ∈ I.
Proposition. A map f : (Dn
, Sn−1
, s0) → (X, A, x0) represents zero in πn(X, A, x0)
iff it is homotopic rel Sn−1
to a map with image in A.
Proof. Suppose that f ∼ g rel Sn−1
and g(Dn
) ⊆ A. Then g = g ◦ idDn is homotopic
to the constant map g ◦ const into x0 ∈ A. Hence [f] = [g] = 0.
Let f be homotopic to the constant map via homotopy h : Dn
× I → X. Have a
look at the picture and consider the subset
C = {(x, t) ∈ Dn
× I; 2 x ≤ 2 − t}
of Dn
× I simultaneously with a vertical retraction r : Dn
× I → C and a horisontal
homeomorphism q : C → Dn
× I.
r q
Dn
× I C Dn
× I
Figure 10.2. Retraction r and homeomorphism q
The maps can be defined in the following way:
r(x, t) =
(x, t) for 2 x ≤ 2 − t,
(x, 2(1 − x ) for 2 x ≥ 2 − t
and
q(x, t) =
2
2 − t
x, t .
69
Now H = h ◦ q ◦ r : Dn
× I → X is a homotopy between H(x, 0) = h(x, 0) = f(x) and
H(x, 1) = g(x) where
g(Dn
) = H(Dn
× I) = h(Dn
× {1} ∪ Sn−1
× I) ⊆ A
and H is a homotopy rel Sn−1
.
A map F : (X, A, x0) → (Y, B, y0) induces again the homomorphism F∗ : πn(X, A, x0)
→ πn(Y, B, y0). Since πn(X, x0, x0) = πn(X, x0) the functor πn on Top∗ can be extended
to a functor from Top2
∗ to Abelian groups Ab for n ≥ 3, to the category of
groups G for n = 2 and to the category Set∗ of sets with distinguished element for
n = 1.
From definitions it is clear that homotopic maps induce the same homomorphisms
between homotopy groups. Hence homotopy equivalent spaces have the same homotopy
groups. Particularly, contractible spaces have trivial homotopy groups.
10.3. Long exact sequence of a pair. Relative homotopy groups fit into the
following long exact sequence of a pair.
Theorem. Let (X, A) be a pair of spaces with a distinguished point x0 ∈ A. Then the
sequence
· · · → πn(A, x0)
i∗
−→ πn(X, x0)
j∗
−→ πn(X, A, x0)
δ
−→ πn−1(A, x0) → . . .
where i : A → X, j : (X, x0) → (X, A) are inclusions and δ comes from restriction,
is exact.
More generally, any triple B ⊆ A ⊆ X induces the long exact sequence
· · · → πn(A, B, x0)
i∗
−→ πn(X, B, x0)
j∗
−→ πn(X, A, x0)
δ
−→ πn−1(A, B, x0) → . . .
Proof. We will prove only the version for the pair (X, A). δ is defined on [f] ∈
πn(X, A, x0) by
δ[f] = [f/In−1
].
Exactness in πn(X, x0). According to the previous proposition j∗i∗ = 0, hence
Im i∗ ⊆ Ker j∗. Let [f] ∈ Ker j∗ for f : (In
, ∂In
) → (X, x0). Using again the previous
proposition f ∼ g rel ∂In
where g : In
→ A. Hence [f] = i∗[g].
Exactness in πn(X, A, x0). δj∗ = 0, hence Im j∗ ⊆ Ker δ. Let [f] ∈ Ker δ, i. e.
f(In
, ∂In
, Jn−1
) → (X, A, x0) and f/In−1
∼ const. Then according to HEP there
is f1 : (In
, ∂In
, Jn
) → (X, x0, x0) homotopic to f. Therefore [f1] ∈ πn(X, x0) and
[f] = j∗[f1].
Exactness in πn(A, x0). Let [F] ∈ πn+1(X, A, x0). Then i ◦ F/In
: In
→ X is a map
homotopic to the constant map to x0 through the homotopy F. (Draw a picture.)
Let f : (In
, ∂In
) → (A, x0) and f ∼ 0 through the homotopy F : In
× I → X such
that F(x, 0) = f(x) ∈ A, F/Jn
= x0. Hence [F] ∈ πn+1(X, A, x0) and δ[F] = [f].
Remark. The boundary operator for a triple (X, A, B) is the composition
πn(X, A)
δ
−→ πn(A)
j∗
−→ πn−1(A, B).
70
10.4. Changing basepoints. Let X be a space and γ : I → X a path connecting
points x0 and x1. This path associates to f : (In
, ∂In
) → (X, x1) a map γ · f :
(In
, ∂In
) → (X, x0) by shrinking the domain of f to a smaller concentric cube in In
and inserting the path γ on each radial segment in the shell between ∂In
and the
smaller cube.
f
γ
γ
γ
γ
x0 x0
x0
x0
x1
Figure 10.3. The action of γ on f
It is not difficult to prove that this assigment has the following properties:
(1) γ · (f + g) ∼ γ · f + γ · g for f, g : (In
, ∂In
) → (X, x1),
(2) (γ + κ) · f ∼ γ · (κ · f) for f : (In
, ∂In
) → (X, x2), γ(0) = x0, γ(1) = x1 = κ(0),
κ(1) = x2.
(3) If γ1, γ2 : I → X are homotopic rel ∂I = {0, 1}, then γ1 · f ∼ γ2 · f.
Hence, every path γ defines an isomorphism
γ : πn(X, γ(1)) → πn(X, γ(0)).
Particulary, we have a left action of the group π1(X, x0) on πn(X, x0).
10.5. Fibrations. Fibration is a dual notion to cofibration. (See 1.7.) It plays an
important role in homotopy theory.
A map p : E → B has the homotopy lifting property, shortly HLP, with respect
to a pair (X, A) if the following commutative diagram can be completed by a map
X × I → E
X × {0} ∪ A × I //
i

E
p

X × I //
77ppppppp
B
A map p : E → B is called a fibration (sometimes also Serre fibration or weak
fibration), if it has the homotopy lifting property with respect to all disks (Dk
, ∅).
Theorem. If p : E → B is a fibration, then it has homotopy lifting property with
respect to all pairs of CW-complexes (X, A).
Proof. The proof can be carried out by induction from (k − 1)-skeleton to k-skeleton
similarly as in the proof of Theorem 2.7 if we show that p : E → B has the homotopy
lifting property with respect to the pair (Dk
, ∂Dk
= Sk−1
). The HLP for this pair
71
follows from the fact that the pair (Dk
× I, Dk
× {0} ∪ Sk−1
× I) is homeomorphic to
the pair (Dk
× I, Dk
× {0}), see the picture below, and the fact that p has homotopy
lifting property with respect to the pair (Dk
, ∅).
Figure 10.4. Homeomorphism (Dn
× I, Dn
× {0} ∪ Sn
× I) → (Dn
×
I, Dn
× {0})
Proposition. Every fibre bundle (E, B, p) is a fibration.
Proof. For the definition of a fibre bundle see 8.1. Let Uα be an open covering of
B with trivializations hα : p−1
(Uα) → Uα × F. We would like to define a lift of a
homotopy G : Ik
× I → B. (We have replaced Dk
by Ik
.) The compactness of Ik
× I
implies the existence of a division
0 = t0 < t1 < · · · < tm = 1, Ij = [tj−1, tj],
such that G(Ij1 ×· · ·×Ijk+1
) lies in some Uα. Now we make a lift H : Ik
×I → E of G,
first on (I1)k+1
and then we add successively the other small cubes. We need retractions
r of cubes C × Ijk+1
= k+1
i=1 Iji
to a suitable part of the boundary C × {0} ∪ A × Ijk+1
where H is already defined. A is a CW-subcomplex of the cube C and we are in the
following situation
C × {0} ∪ A × I
g
//
i

Uα × F
p1

C × I G
//
H
66nnnnnnn
Uα
Now, we can define
H(x, t) = (G(x, t), p2 ◦ g ◦ r)(x, t)
where p2 : Uα × F → F is a projection.
Example. Here you are several examples of fibre bundles.
(1) The projection p : Sn
→ RPn
determines a fibre bundle with the fibre S0
.
(2) The projection p : S2n+1
→ RCn
determines a fibre bundle with the fibre S1
.
(3) The special case is so called Hopf fibration
S1
→ S3
→ CP1
= S2
.
72
(4) Similarly, as complex projective space we can define quaternionic projective space
HPn
. The definition determines the fibre bundle
S3
→ S4n+3
→ HPn
.
(5) The special case of the previous fibre bundle is the second Hopf fibration
S3
→ S7
→ HP1
= S4
.
(6) Similarly, the Cayley numbers enable to define another Hopf fibration
S7
→ S15
→ S8
.
(7) Let H be a Lie subgroup of G. Then we get a fibre bundle given by the projection
p : G → G/H with the fibre H.
(8) Let n ≥ k > l ≥ 1. Then the projection
p : Vn,k → Vn,l, p(v1, v2, . . . , vk) = (v1, v2, . . . , vl)
determines a fibre bundle with the fibre Vn−l,k−l.
(9) Natural projection p : Vn,k → Gn,k is a fibre bundle with the fibre O(k).
10.6. Long exact sequence of a fibration. Consider a fibration p : E → B. Take
a basepoint b0 ∈ B, put F = p−1
(b0) and choose x0 ∈ F.
Lemma. For all n ≥ 1
p∗ : πn(E, F, x0) → πn(B, b0)
is an isomorphism.
Proof. First, we show that p∗ is an epimorphism. Consider f : (In
, ∂In
) → (B, b0).
Let k : Jn−1
→ E be the constant map into x0. Since p is a fibration the commutative
diagram
Jn−1
= In−1
× {1} ∪ ∂In−1
× I
k //

E
p

In−1
× I
f
//
g
55kkkkkkkkk
B
can be completed by g : (In
, ∂In
, Jn−1
) → (E, F, x0). Hence p∗[g] = [f].
Now we prove that p∗ is a monomorphism. Consider f : (In
, ∂In
, Jn−1
) → (E, F, x0)
such that p∗[f] = 0. Then there is a homotopy G : (In
×I, ∂In
×I) → (B, b0) between
pf and the constant map into b0. Denote the constant map into x0 by k. Since p is a
fibration, we complete the following commutative diagram:
Jn−1
× I ∪ In
× {0} ∪ In
× {1}
k∪f∪k
//

E
p

In
× I G
//
H
55kkkkkkkkkk
B
by H : (In
× I, ∂In
× I, Jn−1
× I) → (E, B, x0) which is a homotopy between f and
the constant map k.
73
The notion of exact sequence can be enlarged to groups and also to the category
Set∗ of sets with distinquished elements. Here we have to define Ker f = f−1
(b0) for
f : (A, a0) → (B, b0).
Theorem. If p : E → B be a fibration with a fibre F = p−1(b0), x0 ∈ F and B is path
connected, then the sequence
· · · → πn(F, x0)
i∗
−→ πn(E, x0)
p∗
−→ πn(B, b0)
δ
−→ πn−1(F, x0) → . . .
· · · → π0(F)
i∗
−→ π0(E)
p∗
−→ π0(B).
is exact.
Proof. Substitute the isomorphism p∗ : πn(E, F, x0) → πn(B, b0) into the exact sequence
for the pair (E, F). In this way we get the required exact sequence ending
with
· · · → π0(F, x0) → π0(E, x0).
We can prolong it by one term to the right. The exactness in π0(E, x0) follows from
the fact that every path in B ending in b0 can be lifted to a path in E ending in F.
The direct definition of δ : πn(B, b0) → πn−1(F, x0) is given by
δ[f] = [g/In−1
]
where g is the lift in the diagram
Jn−1 x0
//

E
p

In
f
//
g
<<z
z
z
z
z
B
Some applications of this long exact sequence to computations of homotopy groups
will be given in Section 14.
74
11. Fundamental group
The fundamental group of a space is the first homotopy group. In this section we
describe two basic methods how to compute it.
11.1. Covering space. A covering space of a space X is a space X together with a
map p : X → X such that (X, X, p) is a fibre bundle with a discrete fibre.
In the previous section we have proved that every fibre bundle has homotopy lifting
property with respect to CW-complexes. In the case of covering spaces the lifts of
homotopies are unique:
Proposition. Let p : X → X be a covering space and let Y be a space. Given a
homotopy F : Y × I → X and a map f : Y × {0} → X such that F(−, 0) = pf, there
is a unique homotopy F : Y × I → X making the following diagram commutative:
Y × {0}

f
// X
p

Y × I F
//
F
;;wwwwwwwwww
X
Proof. Since the proof follows the same lines as the proof of the analogous proposition
in 10.5, we outline only the main steps.
(1) Using compactness of I we show that for each y ∈ Y there is a neighbourhood
U such that F can be defined on U × I.
(2) F is uniquely determined on {y} × I for each y ∈ Y .
(3) The lifts of F defined on U1 × I and U2 × I concide on (U1 ∩ U2) × I.
From the uniquiness of lifts of loops and their homotopies starting at a fixed point
we get immediately the following
Corollary. The group homomorphism p∗ : π1(X, x0) → π1(X, x0) induced by a covering
space (X, X, p) is injective. The image subgroup p∗(π1(X, x0)) in π1(X, x0) consists
of loops in X based at x0 whose lifts in X starting at x0 are loops.
11.2. Group actions. A left action of a discrete group G on a space Y is a map
G × Y → Y, (g, y) → g · y
such that 1 · y = y and (g1g2) · y = g1 · (g2 · y). We will call this action properly
discontinuous if each point y ∈ Y has an open neighbourhood U such that g1U ∩g2U =
∅ implies g1 = g2.
An action of a group G on a space Y induces the equivalence x ∼ y if y = g · x for
some g ∈ G. The orbit space Y/G is the factor space Y/ ∼.
A space Y is called simply connected if it is path connected and π1(Y, y0) is trivial
for some (and hence all) base point y0.
75
The following theorem provides a useful method for computation of fundamental
groups.
Theorem. Let Y be a path connected space with a properly discontinuous action of a
group G. Then
(1) The natural projection p : Y → Y/G is a covering space.
(2) G ∼= π1(Y/G, p(y0))/p∗π1(Y, y0). Particularly, if Y is simply connected, then
π1(Y/G) ∼= G.
Proof. Let y ∈ Y and let U be a neighbourhood of y from the definition of properly
discontinuous action. Then p−1
(p(U)) is a disjoint union of gU, g ∈ G. Hence
(Y, Y/G, p) is a fibre bundle with the fibre G.
Applying the long exact sequence of homotopy groups of this fibration we obtain
0 = π1(G, 1) → π1(Y, y0)
p∗
−→ π1(Y/G; p(y0))
δ
−→ π0(G) = G → π0(Y ) = 0.
In general π0 of a fibre is only the set with distinguished point. However, here it has
the group structure given by G. Using the definition of δ from 10.3 one can check that
δ is a group homomorphism. Consequently, the exact sequence implies that
G ∼= π1(Y/G, p(y0))/p∗π1(Y, y0).
Example A. Z acts on real numbers R by addition. The orbit space is R/Z = S1
.
According to the previous theorem
π1(S1
, s) = Z.
The fundamental group of the sphere Sn
with n ≥ 2 is trivial. The reason is that any
loop γ : S1
→ Sn
is homotopic to a loop which is not a map onto Sn
and Sn
without
a point is contractible.
Next, the group Z2 = {1, −1} has an action on Sn
, n ≥ 2 given by (−1) · x = −x.
Hence
π1(RPn
) = Z2.
Example B. The abelian group Z ⊕ Z acts on R2
(m, n) · (x, y) = (x + m, y + n).
The factor R2
/(Z ⊕ Z) is two dimensional torus S1
× S1
. Its fundamental group is
Z ⊕ Z.
Example C. The group G given by two generators α, β and the relation β−1
αβ = α−1
acts on R2
by
α · (x, y) = (x + 1, y), β · (x, y) = (1 − x, y + 1).
The factor R2
/G is the Klein bottle. Hence its fundamental group is G.
76
11.3. Free product of groups. As a set the free product ∗αGα of groups Gα, α ∈ I
is the set of finite sequences g1g2 . . . gm such that 1 = gi ∈ Gαi
, αi = αi+1, called
words. The elements gi are called letters. The group operation is given by
(g1g2 . . . gm) · (h1h2 . . . hn) = (g1g2 . . . gmh1h2 . . . hn)
where we take gmh1 as a single letter gm ·h1 if both elements belong to the same group
Gα. It is easy to show that ∗αGα is a group with the empty word as the identity
element. Moreover, for each β ∈ I there is the natural inclusion iβ : Gβ → ∗αGα.
Up to isomorhism the free product of groups is characterized by the following universal
property: Having a system of group homomorphism hα : Gα → G there is just
one group homomorphism h : ∗αGα → G such that hα = hiα.
Exercise. Describe Z2 ∗ Z2.
11.4. Van Kampen Theorem. Suppose that a space X is a union of path connected
open subsets Uα each of which contains a base point x0 ∈ X. The inclusions
Uα → X induce homomorphisms jα : π1(Uα) → π1(X) which determine a unique
homomorphism ϕ : ∗απ1(Uα) → π1(X).
Next, the inclusions Uα ∩ Uβ → Uα induce the homomorphisms iαβ : π1(Uα ∩ Uβ) →
π1(Uα). We have jαiαβ = jβiβα. Consequently, the kernel of ϕ contains elements of
the form iαβ(ω)iβα(ω−1
) for any ω ∈ π1(Uα ∩ Uβ).
Van Kampen Theorem provides the full description of the homomorphism ϕ which
enables us to compute π1(X) using groups π1(Uα) and π1(Uα ∩ Uβ).
Theorem (Van Kampen Theorem). If X is a union of path connected open sets Uα
each containing a base point x0 ∈ X and if each intersection Uα ∩Uβ is path connected,
then the homomorhism ϕ : ∗απ1(Uα) → π1(X) is surjective. If in addition each intersection
Uα ∩ Uβ ∩ Uγ is path connected, then the kernel of ϕ is the normal subgroup
N in ∗απ1(Uα) generated by elements iαβ(ω)iβα(ω−1
) for any ω ∈ π1(Uα ∩ Uβ). So ϕ
induces an isomorphism
π1(X) ∼= ∗απ1(Uα)/N.
Example. If Xα are path connected spaces, then
π1( Xα) = ∗απ1(Xα).
Outline of the proof of Van Kampen Theorem. For simplicity we suppose that X is a
union of only two open subsets U1 and U2.
Surjectivity of ϕ. Let f : I → X be a loop starting at x0 ∈ U1 ∪ U2. This loop
is up to homotopy a composition of several paths, for simplicity suppose there are
three such that f1 : I → U1, f2 : I → U2 and f3 : I → U1 with end points succesively
x0, x1, x2, x0 ∈ U1 ∩U2. Since U1 ∩U2 is path connected there are paths g1 : I → U1 ∩U2
and g2 : I → U1 ∩ U2 from x0 to x1 and x2, respectively. Then the loop f is up to
homotopy the composition of loops f1 − g1 : I → U1, g1 + f2 − g2 : I → U2 and
g2 + f3 : I → U1. Consequently, [f] ∈ π1(X) lies in the image of ϕ.
Kernel of ϕ. Suppose that the image under ϕ of a word with m letters [f1][g1][f2] . . . ,
where [fi] ∈ π1(U1), [gi] ∈ π1(U2), is zero in π1(X). Then there is a homotopy
77
f2
f1
f3
x0
x1
x2
g1
g2
Figure 11.1. [f] = [f1 + f2 + f3] = [f1 − g1] + [g1 + f2 − g2] + [g2 + f3]
F : I × I → X such that
F(s, 0) = f1 + g1 + f2 + . . . , F(s, 1) = x0, F(0, t) = F(1, t) = x0
where we suppose that fi is defined on [2i−2
m
, 2i−1
m
] and gi is defined on [2i−1
m
, 2i
m
]. Since
I × I is compact, there is an integer n, a multiple of m, such that
F
i
n
,
i + 1
n
×
j
n
,
j + 1
n
is a subset in U1 or U2. Using homotopy extension property, we can construct a
homotopy from F to F rel J1
such that again
F
i
n
,
i + 1
n
×
j
n
,
j + 1
n
is a subset in U1 or U2, and moreover,
F
i
n
,
j
n
= x0.
Further, F(s, 0) = f 1 + g 1 + f 2 + . . . where f i ∼ fi, g i ∼ gi in U1 and U2, respectively,
rel the boundary of the domain of definition. We want to show that the word
[f 1]1[g 1]2[f 2]1 . . . belongs to N. Here [ ]i stands for an element in π1(Ui).
We can decompose
I × I =
i
Mi
where Mi is a maximal subset with the properties:
(1) Mi is a union of several squares [ i
n
, i+1
n
] × [ j
n
, j+1
n
].
(2) int Mi is path connected.
(3) F(Mi) is a subset in U1 or U2.
78
For simplicity suppose that we have four sets Mi as indicated in the picture.
x0
x0 x0
M1
M2
M3
M4
p
k
l
f1 g1 f2
Figure 11.2. [f1]1[g1]2[f2]1 ∈ Kerϕ
In this situation there are three loops k, l and p starting at x0 and lying in U1 ∩ U2.
They are defined by F on common boundary of M1 and M2, M2 and M3, M3 and M4,
respectively. Now, we get
[f 1]1[g 1]2[f 2]1 = [k]1[−k + l]2[−l + p]1 = [k]1[−k]2[l]2[−l]1[p]1
= [k]1[−k]2[l]2[−l]1 ∈ N.
Corollary. Let X be a union of two open subsets U and V where V is simply connected
and U ∩ V is path connected. Then
π1(X) = π1(U)/N
where N is the normal subgroup in π1(U) generated by the image of π1(U ∩ V ).
Exercise. Use the previous statement to compute the fundamental group of the Klein
bottle and other 2-dimensional closed surfaces.
11.5. Fundamental group and homology. Here we compare the fundamental
group of a space with the first homology group. We obtain a special case of Hurewitz
theorem, see 13.6.
Theorem. By regarding loops as 1-cycles, we obtain a homomorphism h : π1(X, x0) →
H1(X). If X is path connected, then h is surjective and its kernel is the commutator
subgroup of π1(X). So h induces isomorphism from the abelization of π1(X, x0) to
H1(X).
For the proof we refer to [Hatcher], Theorem 2A.1, pages 166–167.
79
12. Homotopy and CW-complexes
This section demonstrates the importance of CW-complexes in homotopy theory.
The main results derived here are Whitehead theorem and theorems on approximation
of maps by cellular maps and spaces by CW-complexes.
12.1. n-connectivity. A space X is n-connected if πi(X, x0) = 0 for all 0 ≤ i ≤ n
and some base point x0 ∈ X (and consequently, for all base points).
A pair (X, A) is called n-connected if each component of path connectivity of X
contains a point from A and πi(X, A, x0) = 0 for all x0 ∈ A and all 1 ≤ i ≤ n
We say that a map f : X → Y is an n-equivalence if f∗ : πi(X, x0) → πi(Y, f(x0)) is
an isomorphism for all x0 ∈ X if 0 ≤ i < n and an epimorphism for all x0 if i = n.
Exercise. Prove that a pair (X, A) is n-connected if and only if the inclusion i : A →
X is an n-equivalence.
12.2. Compression lemma is an important technical tool in what follows.
Lemma A (Compression lemma). Let (X, A) be a pair of CW-complexes and (Y, B)
a pair with B = ∅. Suppose that πn(Y, B, y0) = 0 for all y0 ∈ B whenever there is a
cell in X − A of dimension n. Then every f : (X, A) → (Y, B) is homotopic rel A
with a map g : X → B.
A

f/A
// B

X
f
//
g
∼
>>~
~
~
~
Y
If n = 0, the condition π0(Y, B, y0) = 0 means that (Y, B) is 0-connected.
Proof. By induction we will define maps fn : X → Y such that fn(Xn
∪ A) ⊆ B,
and fn is homotopic to fn−1 rel A ∪ Xn−1
. Put f−1 = f. Suppose that we have
fn−1 and there is a cell en
in X − A. Let ϕ : Dn
→ X be its characteristic map.
Then fn−1ϕ : (Dn
, ∂Dn
) → (Y, B) represents zero element in πn(Y, B). According to
Proposition 10.2 it means that fn−1ϕ : (Dn
, ∂Dn
) → (Y, B) is homotopic rel ∂Dn
to
a map hn : (Dn
, ∂Dn
) → (B, B). Doing it for all cells of dimension n in X − A we
obtain a map gn : Xn
∪A → B homotopic rel A∪Xn−1
with fn−1 restricted to Xn
∪A.
Using the homotopy extension property of the pair (X, Xn
∪ A) we can conclude that
gn can be extended to a map fn : X → Y which is homotopic rel A ∪ Xn−1
to fn−1.
Now for x ∈ Xn
define g(x) = fn(x) = gn(x). By the same trick as in the proof of
Theorem 2.7 we can construct a homotopy rel A between f and g.
The proof of the following extension lemma is similar but easier and hence left to
the reader.
Lemma B (Extension lemma). Consider a pair (X, A) of CW-complexes and a map
f : A → Y . If Y is path connected and πn−1(Y, y0) = 0 whenever there is a cell in
X − A of dimension n, then f can be extended to a map X → Y .
80
12.3. Whitehead Theorem. The compression lemma has two important conse-
quences.
Corollary. Let h : Z → Y be an n-equivalence and let X be a finite dimensional
CW-complex. Then the induced map h∗ : [X, Z] → [X, Y ] is
(1) a surjection if dim X ≤ n,
(2) a bijection if dim X ≤ n − 1.
Proof. First, we will suppose that h : Z → Y is an inclusion and apply the compression
lemma. Put B = Z, A = ∅ and consider a map f : X → Y . If dim X ≤ n then all
the assumptions of the compression lemma are satisfied. Consequently, there is a map
g : X → Z such that hg ∼ f. Hence h∗ : [X, Z] → [X, Y ] is surjection.
Let dim X ≤ n − 1 and let g1, g2 : X → Z be two maps such that hg1 ∼ hg2 via a
homotopy F : X ×I → Y . Then we can apply the compression lemma in the situation
of the diagram
X × {0, 1}
g1∪g2
//

Z
h

X × I F
//
H
::t
t
t
t
t
Y
to get a homotopy H : X × I → Z between g1 and g2.
If h is not an inclusion, we use the mapping cylinder Mh. (See 1.5 for the definition
and basic properties.) Let f : X → Y be a map. Apply the result of the previous part
of the proof to the inclusion iZ : Z → Mh and to the map iY f : X → Y → Mh to get
g : X → Z such that iZg ∼ iY f.
Z
h
~~||||||||
iZ

h
BBBBBBBB
X
g
77nnnnnnnn
f
// Y iY
// Mh p
// Y
Since the right triangle in the diagram commutes and the middle one commutes up to
homotopy and piY = idY, we get
hg = piZg ∼ piY f = f.
The statement (2) can be proved in a similar way.
A map f : X → Y is called a weak homotopy equivalence if f∗ : πn(X, x0) →
πn(Y, f(x0)) is an isomorphism for all n and all base points x0.
Theorem (Whitehead Theorem). If a map h : Z → Y between two CW-complexes is
a weak homotopy equivalence, then h is a homotopy equivalence.
Moreover, if Z is a subcomplex of Y and h is an inclusion, then Z is even deformation
retract of Y .
81
Proof. Let h be an inclusion. We apply the compression lemma in the following situ-
ation:
Z
idZ
//
h

Z
h

Y
idY
//
g
??~
~
~
~
Y
Then gh ∼ idY rel Z and consequently hg = idZ. So Z is a deformation retract of Y .
The proof in a general case again uses mapping cylinder Mh.
12.4. Simplicial approximation lemma. The following rather technical statement
will play an important role in proofs of approximation theorems in this section and in
the proof of homotopy excision theorem in the next section. Under convex polyhedron
we mean an intersection of finite number of halfspaces in Rn
with nonempty interior.
Lemma (Simplicial approximation lemma). Consider a map f : In
→ Z. Let Z be a
space obtained from a space W by attaching a cell ek
. Then f is rel f−1
(W) homotopic
to f1 for which there is a simplex ∆k
⊂ ek
with f−1
1 (∆k
) a union (possibly empty) of
finitely many convex polyhedra such that f1 is the restriction of a linear surjection
Rn
→ Rk
on each of them.
The proof is elementary but rather technical and we omit it. See [Hatcher], Lemma
4.10, pages 350–351.
12.5. Cellular approximation. We recall that a map g : X → Y between two
CW-complexes is called cellular, if g(Xn
) ⊆ Y n
for all n.
Theorem (Cellular approximation theorem). If f : X → Y is a map between CWcomplexes,
then it is homotopic to a cellular map. If f is already cellular on a subcomplex
A, then f is homotopic to a cellular map rel A.
Corollary A. πk(Sn
) = 0 for k < n.
Corollary B. Let (X, A) be a pair of CW-complexes such that X − A contains only
cells of dimension greater then n. Then (X, A) is n-connected.
Proof of the cellular approximation theorem. By induction we will construct maps fn :
X → Y such that f−1 = f, fn is cellular on Xn
and fn ∼ fn−1 rel Xn−1
∪ A. Then we
can define g(x) = fn(x) for x ∈ Xn
and by the same trick as in the proof of Theorem
2.7 we can construct homotopy rel A between f and g.
Suppose we have already fn−1 and there is a cell en
such that fn−1(en
) does not lie in
Y n
. Then f(en
) meets a cell ek
in Y of dimension k > n. According to the simplicial
approximation lemma fn−1 restricted to en is homotopic rel ∂en
to h : en → Y with
the property that there is a simplex ∆k
⊂ ek
and h(en
) ⊂ Y − ∆k
. (Since n < k,
there is no linear surjection Rn
→ Rk
.) ∂ek
is a deformation retract of ek − ∆k
and
that is why h is homotopic rel ∂en
to a map g : en → Y − ek
. Since f(en
) meets only
a finite number of cells, repeating the previous step we get a map fn defined on en
82
such that fn(en
) ⊆ Y n
and homotopic rel ∂en
to fn−1/ en. In the same way we can
define fn on A ∪ Xn
homotopic to fn−1/A ∪ Xn
rel A ∪ Xn−1
. Then using homotopy
extension property for the pair (X, A ∪ Xn
) we obtain fn : X → Y homotopic to fn−1
rel A ∪ Xn−1
.
12.6. Approximation by CW-complexes. Consider a pair (X, A) where A is a
CW-complex. An n-connected CW model for (X, A) is an n-connected pair of CWcomplexes
(Z, A) together with a map f : Z → X such that f/A = idA and f∗ :
πi(Z, z0) → πi(X, f(z0)) is an isomorphism for i > n and a monomorphism for i = n
and all base points z0 ∈ Z.
If we take A a set containing one point from every path component of X, then
0-connected CW model gives a CW-complex Z and a map Z → X which is a weak
homotopy equivalence.
Theorem A (CW approximation theorem). For every n ≥ 0 and for every pair
(X, A) where A is a CW-complex there exists n-connected CW-model (Z, A) with the
additional property that Z can be obtained from A by attaching cells of dimensions
greater than n.
Proof. We proceed by induction constructing Zn = A ⊂ Zn+1 ⊂ Zn+2 ⊂ . . . with Zk
obtained from Zk−1 by attaching cells of dimension k, and a map f : Zk → X such
that f/A = idA and f∗ : πi(Zk) → πi(X) is a monomorhism for n ≤ i < k and an
epimorphism for n < i ≤ k. For simplicity we will consider X and A path connected
with a fixed base point x0 ∈ A.
Suppose we have already f : Zk → X. Let ϕα : Sk
→ Zk be maps representing
generators in the kernel of f∗ : πk(Zk) → πk(X). Put
Yk+1 = Zk ∪ϕα
α
Dk+1
α .
Since the map f : Zk → X restricted to the boundaries of new cells is trivial, it can
be extended to a map f : Yk+1 → X.
By the cellular approximation theorem πi(Yk+1) = πi(Zk) for all i ≤ k − 1. Hence
the new f∗ has the same properties as the old f∗ on homotopy groups πi with i ≤
k − 1. Since the composion πk(Zk) → πk(Yk+1) → πk(X) is surjective according to the
induction assumptions, the homomorphism f∗ : πk(Yk+1) → πk(X) has to be surjective
as well.
Now we prove that it is injective. Let [ϕ] ∈ πk(Yk+1) and let fϕ ∼ 0. By cellular
approximation ϕ : Sk
→ Yk+1 is homotopic to ϕ : Sk
→ Y k
k+1 = Zk ⊆ Yk+1 and
[fϕ] = 0 in πk(X). Hence [ϕ] ∈ Ker f∗ is a sum of [ϕα], and consequenly, it is zero in
πk(Yk+1).
Next, let maps ψα : Sk+1
α → X represent generators of πk+1(X). Put
Zk+1 = Yk+1 ∨
α
Sk+1
α
and define f = ψα on new (k + 1)-cells. It is clear that f∗ : πk+1(Zk+1) → πk+1(X)
is a surjection. Using cellular approximation it can be shown that πi(Zk+1, Yk+1) =
83
0 for i ≤ k. From the long exact sequence of the pair (Zk+1, Yk+1) we get that
πi(Yk+1) = πi(Zk+1) for i ≤ k − 1. Consequently, f∗ : πi(Zk+1) → πi(X) is an
isomorphism for n < i ≤ k − 1 and a monomorphism for i = n. The same long exact
sequence implies that πk(Yk+1) → πk(Zk+1) is surjective. We have already proved that
f∗ : πk(Yk+1) → πk(X) is an isomorphism. From the diagram
πk(Yk+1)
iso &&MMMMMMMMMM
epi
// πk(Zk+1)
f∗

πk(X)
we can see that f∗ : πk(Zk+1) → πk(X) is also an isomorphism.
Corollary. If (X, A) is an n-connected pair of CW-complexes, then there is a pair
(Z, A) homotopy equivalent to (X, A) rel A such that the cells in Z −A have dimension
greater than n.
Proof. Let f : (Z, A) → (X, A) be an n-connected model for (X, A) obtained by
attaching cells of dimension > n to A. Then f∗ : πj(Z) → πj(X) is a monomorphism
for j = n and an isomorphism for j > n. We will show that f∗ is an isomorphism also
for j ≤ n. Consider the diagram:
A
iZ

iX
@@@@@@@@
Z
f
// X
The inclusions iX and iZ are n-equivalences. Consequently, f∗iZ∗ = iX∗ : πj(A) →
πj(X) is an epimorphism for j = n. Hence so is f∗. Next, iX∗ and iZ∗ are isomorphisms
for j < n, hence so is f∗.
Finally, according to Whitehead Theorem, the weak homotopy equivalence f between
two CW-complexes is a homotopy equivalence.
Theorem B. Let f : (Z, A) → (X, A) and f : (Z , A ) → (X , Z ) be two n-connected
CW-models. Given a map g : (X, A) → (X , A ) there is a map h : (Z, A) → (Z , A )
such that the following diagram commutes up to homotopy rel A:
Z
f
//
h

X
g

Z
f
// X
The map h is unique up to homotopy rel A.
84
Proof. By the previous corollary we can suppose that Z −A has only cells of dimension
≥ n + 1. We can define h/A as g/A.
A
h/A
//

Z
f

Z
gf
// X
Replace X by the mapping cylinder Mf which is homotopy equivalent to X . Since
f : Z → X is an n-connected model, from the long exact sequence of the pair
(Mf , Z ) we get that πi(Mf , Z ) = 0 for i ≥ n + 1. According to compression lemma
12.2 there exists h : Z → Z such that the diagram
A
h/A
//

Z

Z //
h
>>|
|
|
|
|
Mf
commutes up to homotopy rel A. This h has required properties. The proof that it is
unique up to homotopy follows the same lines.
85
13. Homotopy excision and Hurewicz theorem
One of the reasons why the computation of homotopy groups is so difficult is the
fact that we have no general excision theorem at our disposal. Nevertheless, there is
a restricted version of such a theorem. It has many consequences, one of them is the
Freudenthal suspension theorem which enables us to compute πn(Sn
). At the end of
this section we define the Hurewicz homomomorphism which under certain conditions
compares homotopy and homology groups.
13.1. Homotopy excision theorem. Excision theorem for homology groups has
the following restricted analogue for homotopy groups.
Theorem (Blakers-Massey theorem). Let A and B be subcomplexes of CW-complex
X = A ∪ B. Suppose that C = A ∩ B is connected, (A, C) is m-connected and (B, C)
is n-connected. Then the inclusion
j : (A, C) → (X, B)
is (m+n)-equivalence, i. e. j∗ : πi(A, C) → πi(X, B) is an isomorphism for i < m+n
and an epimorphism for i = m + n.
Proof. We distinguish several cases.
1. Suppose that A = C ∪ α em+1
α and B = C ∪ en+1
. First we prove that j∗ :
πi(A, C) → πi(X, B) is surjective for i ≤ m + n.
Consider f : (Ii
, ∂Ii
, Ji−1
) → (X, B, x0). Using simplicial approximation lemma
12.4 we can suppose that there are simplices ∆m+1
α ⊂ em+1
α and ∆n+1
⊂ en+1
such that
their inverse images f−1
(∆m+1
α ), f−1
(∆n+1
) are unions of convex polyhedra on each of
which f is a linear surjection Ri
onto Rm+1
and Rn+1
, respectively. We will need the
following statement.
Lemma. If i ≤ m + n then there exist points pα ∈ ∆m+1
α , q ∈ ∆n+1
and a continuous
function ϕ : Ii−1
→ [0, 1) such that
(a) f−1
(pα) lies above the graph of ϕ,
(b) f−1
(q) lies below the graph of ϕ,
(c) ϕ = 0 on ∂Ii−1
.
Let us postpone the proof of the lemma for a moment. The subspace M = {(s, t) ∈
Ii−1
× I; t ≥ ϕ(s)} is a deformation retract of Ii
with deformation retraction h :
Ii
× I → Ii
, h(x, 0) = x, h(x, 1) ∈ M. Then
H = fh : Ii
× I → X
provides a homotopy between f and
g : (Ii
, ∂Ii
, Ji−1
) → (X − {q}, X − {q} − {pα}, x0).
Obviously, g is homotopic to ˜g : (Ii
, ∂Ii
, Ji−1
) → (A, C, x0). Hence j∗[˜g] = [f].
The fact that j∗ : πi(A, C) → πi(X, B) is monomorphism for i ≤ m + n − 1 can
be proved by the same way as above replacing f by homotopy h : Ii
× I → (X, B).
(Notice that i + 1 ≤ m + n now.)
86
1
0
f−1
(pα)
f−1
(q)
ϕ
π
Ii−1
Figure 13.1. The graph of ϕ
Proof of the lemma. Choose arbitrary q ∈ ∆n+1
. Then f−1
(q) is a union of convex
simplices of dimension ≤ i − n − 1. Denote π : Ii
→ Ii−1
the projection given
by omitting the last coordinate. π−1
(π(f−1
(q))) is the union of convex simplices of
dimension ≤ i − n. On the set π−1
(π(f−1
(q))) ∩ f−1
(∆m+1
α ) is f linear, hence
f(π−1
(π(f−1
(q)))) ∩ ∆m+1
α
is the union of simplices of dimension at most i−n < m+1 for i ≤ m+n. Consequently,
there is pα ∈ ∆m+1
α such that
f−1
(pα) ∩ π−1
(πf−1
(q)) = ∅.
Since Im f meets only finite number of cells em+1
α , the set π(f−1
(pα)) is compact and
disjoint from π(f−1
(q)). Hence there is continuous function ϕ, ϕ = 0 on π(f−1
(pα))
and ϕ = 1 − ε on π(f−1
(q)) with required properties.
2. Suppose that A is obtained from C by attaching cells em+1
α and B is obtained
by attaching cells e
nβ
β of dimensions ≥ n + 1. Consider a map f : (Ii
, ∂Ii
, Ji−1
) →
(X, B, x0). f meets only finite number of cells e
nβ
β . According to the case 1 we can
show that f is homotopic to
f1 :(Ii
, ∂Ii
) → (X − en1
, B − en1
),
f2 :(Ii
, ∂Ii
) → (X − en1
− en2
, B − en1
− en2
),
. . .
fr :(Ii
, ∂Ii
) → (A, C).
3. Suppose that A is obtained from C by attaching cells of dimensions ≥ m + 1
and B is obtained by attaching cells of dimensions ≥ n + 1. We may assume that
the dimensions of new cells in A is ≤ m + n + 1 since higher dimensional ones have
no effect on πi for i ≤ m + n by cellular approximation theorem 12.5. Let Ak be a
CW-subcomplex of A obtained from C by attaching cells of dimension ≤ k, similarly
let Xk be a CW-subcomplex of X obtained from B by attaching cells of dimension
≤ k. Using the long exact sequences for triples (Ak, Ak−1, C) and (Xk, Xk−1, B), we
87
get the diagram
πi+1(Ak, Ak−1) //
∼=

πi(Ak−1, C) //

πi(Ak, C) //

πi(Ak, Ak−1) //
∼=

πi−1(Ak−1, C)

πi+1(Xk, Xk−1) // πi(Xk−1, B) // πi(Xk, B) // πi(Xk, Xk−1) // πi−1(Xk−1, B)
Applying the previous step for Xk = Ak ∪ Xk−1 and Ak−1 = Ak ∩ Xk−1 we obtain the
indicated isomorphisms. Now the induction with respect to k and 5-lemma completes
the proof that πi(Am+n+1, C) → πi(Xm+n+1, B) is an isomorphism for i < m + n and
an epimorphism for i = m + n.
4. Consider a general case. Then according to Corrolary 12.6 there is a CW-pair
(A , C) homotopy equivalent to (A, C) and a CW-pair (B , C) homotopy equivalent
to (B, C) such that A − C contains only cells of dimension ≥ m + 1 and B − C
contains only cells of dimension ≥ n + 1. Then X = A ∪ B is homotopy equivalent
to X = A ∪ B. According to the previous case j : (A , C) → (X , B ) is an (m + n)equivalence,
consequently j : (A, C) → (X, B) is an (m + n)-equivalence as well.
Corollary. If a CW-pair (X, A) is r-connected and A is s-connected with r, s ≥ 0,
then the homomorphism
πi(X, A) → πi(X/A)
induced by the quotient map X → X/A is an isomorphism for i ≤ r + s and an
epimorphism for i ≤ r + s + 1.
Proof. Consider the diagram:
πi(X, A) // πi(X ∪ CA, CA)

// πi(X ∪ CA/CA)
∼= // πi(X/A)
πi(X ∪ CA)
∼=
OO
∼=
55kkkkkkkkkkkkkk
The first homomorphism is (r+s+1)-equivalence by the homotopy excision theorem for
(s+1)-connected pair (CA, A) and r-connected pair (X, A). The vertical isomorphism
comes from the long exact sequence for the pair (X ∪ CA, CA) and the remaining
isomorphisms are induced by a homotopy equivalence and the identity X ∪CA/CA =
X/A.
13.2. Freudenthal suspension theorem. We have defined the suspension of a
space in 1.5 and the reduced suspension of a space with distinquished point in 1.6. In
4.3 we have introduced the suspension of a map. In a similar way we can define the
reduced suspension of a map which preserves distinquished points. This notion defines
so called suspension homomorphism πi(X) → πi+1(X), [f] → [Σf] for every space X.
Theorem (Freudenthal suspension theorem). Let X be (n−1)-connected CW-complex,
n ≥ 1. Then the suspension homomorphism πi(X) → πi+1(ΣX) is an isomorphism
for i ≤ 2n − 2 and an epimorphism for i ≤ 2n − 1.
88
Proof. The suspension ΣX is a union of two reduced cones C+X and C−X with
intersection X. Now, we get
πi(X) ∼= πi+1(C+X, X) → πi+1(ΣX, C−X) ∼= πi+1(ΣX)
where the first and the last isomorphisms come from the long exact sequences for pairs
(C+X, X) and (ΣX, C−X), respectively, and the middle homomorphism comes from
homotopy excision theorem for n-connected pairs (C+X, X) and (C−X, X). What
remains is to show that the induced map on the level of homotopy groups is the same
as suspension homomorphism which is left to the reader.
13.3. Stable homotopy groups. The Freudenthal suspension theorem enables us
to define stable homotopy groups. Consider a based space X and an integer j. The
n-times iterated reduced suspension Σn
X is at least (n − 1)-connected. If n ≥ j + 2,
then i = j + n ≤ 2n − 2, so the assumptions of the Freudenthal suspension theorem
are satisfied and we get
πj+(j+2)(Σj+2
X) ∼= πj+(j+3)(Σj+3
X) ∼= πj+(j+4)(Σj+4
X) ∼= . . .
Hence we define the j-th stable homotopy group of the space X as
πs
j (X) = lim
n→∞
πj+n(Σn
X).
We will write πs
j for the j-th stable homotopy group of S0
.
13.4. Computations. In this paragraph we compute n-th homotopy groups of (n −
1)-connected CW-complexes.
Theorem A. πn(Sn
) ∼= Z generated by the identity map for all n ≥ 1. Moreover, this
isomorphism is given by the degree map πn(Sn
) → Z.
Proof. Consider the diagram
π1(S1
)
epi
//
∼=deg

π2(S2
)
∼= //
deg

π3(S3
)
∼= //
deg

. . .
Z
= // Z
= // Z
= // . . .
where the horizontal homomorphisms are suspension homomorphisms and the left vertical
isomorphism is known from Section 11 and determined by degree. The statement
follows now from the fact that deg f = deg Σf.
Exercise. Prove that πn( α∈A Xα) = α∈A πn(Xα).
Theorem B. πn( α∈A Sn
α) = α∈A Z for n ≥ 2.
Proof. Suppose first that A is finite. Then CW-complex α∈A Sn
α is a subcomplex of
CW-complex α∈A Sn
α. The pair
α∈A
Sn
α,
α∈A
Sn
α
89
is (2n − 1)-connected since α∈A Sn
α is obtained from α∈A Sn
α by attaching cells of
dimension ≥ 2n. Hence
πn(
α∈A
Sn
α) = πn(
α∈A
Sn
α) =
α∈A
πn(Sn
α) =
α∈A
πn(Sn
α) =
α∈A
Z.
If A is infinite, consider homomorphism φ : α∈A πn(Sn
α) → πn( α∈A Sn
α) induced
by inclusions πn(Sn
α) → α∈A Sn
α. φ is surjective since any f : Sn
→ α∈A Sn
α has a
compact image and meets only finitely many Sn
α’s. Similarly, if h : Sn
× I → α∈A Sn
α
is homotopy between f and the constant map, it meets only finitely many Sn
α’s, so
φ−1
([f]) is zero.
Theorem C. Suppose n ≥ 2. If X is obtained from α∈A Sn
α by attaching cells en+1
β
via base point preserving maps ϕβ : Sn
→ α∈A Sn
α, then
πi(X) =
0 if i < n,
α∈A πn(Sn
α)/N if i = n.
where N is a subgroup of α∈A πn(Sn
α) generated by [ϕβ].
Proof. The first equality is clear from the cellular approximation theorem. Consider
the long exact sequence for the pair (X, Xn
= α∈A Sn
α)
πn+1(X, Xn
)
∂
−→ πn(Xn
) → πn(X) → 0.
The pair (X, Xn
) is n-connected, Xn
is (n − 1)-connected, hence by Corollary 13.1
πn+1(X, Xn
) → πn+1(X/Xn
) = πn+1(
β∈B
Sn+1
β ) =
β∈B
Z
is an isomorphism. Hence
πn(X) = πn(Xn
)/ Im ∂ = πn(
α∈A
Sn
α)/N
since Im ∂ is generated by [ϕβ].
13.5. Hurewicz homomorphism. The Hurewicz map h : πn(X, A, x0) → Hn(X, A)
assigns to every element in πn(X, A, x0) represented by f : (Dn
, ∂Dn
, s0) → (X, A, x0)
the element f∗(ι) ∈ Hn(X, A) where ι ∈ Hn(Dn
, ∂Dn
) = Hn(∆n
, ∂∆n
) is the generator
induced by the identity map ∆n
→ ∆n
. In the same way we can define the Hurewicz
map h : πn(X) → Hn(X).
Proposition 13.6. The Hurewicz map is a homomorphism.
Proof. Let c : Dn
→ Dn
∨ Dn
be the map collapsing equatorial Dn−1
into a point,
q1, q2 : Dn
∨ Dn
→ Dn
quotient maps and i1, i2 : Dn
→ Dn
∨ Dn
inclusions. We have
90
the diagram
Hn(Dn
, ∂Dn
)
c∗
// Hn(Dn
∨ Dn
, ∂Dn
∨ ∂Dn
)
f∨g
//
q1∗⊕q2∗

Hn(X, A)
Hn(Dn
, ∂Dn
) ⊕ Hn(Dn
, ∂Dn
)
i1∗+i2∗
OO
Since i1∗ + i2∗ is an inverse to q1∗ ⊕ q2∗, we get
h([f] + [g]) = (f + g)∗(ι) = (f ∨ g)∗c∗(ι)
= (f ∨ g)∗(i1∗ + i2∗) (q1∗ ⊕ q2∗)c∗ (ι) = (f∗ + g∗)(ι ⊕ ι)
= f∗(ι) + g∗(ι) = h([f]) + h([g]).
We leave the reader to prove the following properties of the Hurewicz homomorphism
directly from the definition:
Proposition 13.7. The Hurewicz homomorphism is natural, i. e. the diagram
πn(X, A)
f∗
//
hX

πn(Y, B)
hY

Hn(X, A)
f∗
// Hn(Y, B)
commutes for any f : (X, A) → (Y, B).
The Hurewicz homomorphisms make commutative also the following diagram with
long exact sequences of a pair (X, A):
πn(A) //
hA

πn(X) //
hX

πn(X, A)
∂ //
h(X,A)

πn−1(A)
hA

Hn(A) // Hn(X) // Hn(X, A)
∂ // Hn−1(A)
13.8. Hurewicz theorem. The previous calculations of πn( α∈A Sn
α) enable us to
compare homotopy and homology groups of (n − 1)-connected CW-complexes via the
Hurewicz homomorphism.
Theorem A (Absolute version of the Hurewicz theorem). Let n ≥ 2. If X is a (n−1)connected,
then ˜Hi(X) = 0 for i < n and h : πn(X) → Hn(X) is an isomorphism.
For the case n = 1 see Theorem 11.5.
Proof. We will carry out the proof only for CW-complexes X. For general method
which enables us to enlarge the result to all spaces see [Hatcher], Proposition 4.21.
First, realize that h : πn(Sn
) → Hn(Sn
) is an isomorphism. It follows from the
characterization of πn(Sn
) by degree in Theorem 13.4.
According to Corollary 12.6 every (n − 1)-connected CW-complex X is homotopy
equivalent to a CW-complex obtained by attaching cells of dimension ≥ n to a point.
91
Moreover cells of dimension ≥ n + 2 do not play any role in computing πi and Hi for
i ≤ n. Hence we may suppose that
X =
α∈A
Sn
α ∪ϕβ
β∈B
en+1
β = Xn+1
where ϕβ are base point preserving maps. Then ˜Hi(X) = 0 for i < n.
Using the long exact sequences for the pair (X, Xn
) and the Hurewicz homomorphisms
between them we get
πn+1(X, Xn
)
∂ //
h

πn(Xn
) //
h

πn(X) //
h

0
Hn+1(X, Xn
)
∂ // Hn(Xn
) // Hn(X) // 0
Since πn+1(X, Xn
) is isomorphic to πn+1(X/Xn
) = πn+1(Sn+1
β ) and πn(Xn
) =
πn(Sn
α), the first and the second Hurewicz homomorphisms are isomorphisms. According
to the 5-lemma so is h : πn(X) → Hn(X).
Let [γ] ∈ π1(A, x0), [f] ∈ πn(X, A, x0). Then γ · f and f are homotopic (although
the homotopy does not keep the base point x0 fixed), and consequently,
(γ · f)∗(ι) = f∗(ι)
for ι ∈ Hn(Dn
, ∂Dn
). Hence h([γ] · [f]) = h([f]).
Let πn(X, A, x0) be the factor of πn(X, A, x0) by the normal subgroup generated by
[γ] · [f] − [f]. Let h : πn(X, A, x0) → Hn(X, A) be the map induced by the Hurewicz
homomorphism h.
Theorem B (Relative version of the Hurewicz theorem). Let n ≥ 2. If a pair (X,A)
of the path connected spaces is (n − 1)-connected, then Hi(X, A) = 0 for i < n and
h : πn(X, A, x0) → Hn(X, A) is an isomorphism.
Proof. We will prove the theorem for a pair (X, A) of CW-complexes where A is
supposed to be simply connected. In this case πn(X, A, x0) = πn(X, A, x0) and h = h.
For general proof see [Hatcher], Theorem 4.37, pages 371–373.
Since (X, A) is (n − 1)-connected and A is 1-connected, Corollary 13.1 implies
that the quotient map πn(X, A) → πn(X/A) is an isomorphism and X/A is (n − 1)connected.
The absolute version of the Hurewicz theorem and the commutativity of
the diagram
πn(X, A)
∼= //
h

πn(X/A)
h∼=

Hn(X, A)
∼= // Hn(X/A)
imply immediately the required statement.
92
13.9. Homology version of Whitehead theorem. Since computations in homology
are much easier that in homotopy, the following homology version of the Whitehead
theorem gives a very useful method how to prove that two spaces are homotopy
equivalent.
Theorem (Whitehead theorem). A map f : X → Y between two simply connected
CW-complexes is homotopy equivalence if f∗ : Hn(X) → Hn(Y ) is an isomorphism for
all n.
Proof. Replacing Y by the mapping cylinder Mf we can consider f to be an inclusion
X → Y . Since X and Y are simply connected, we have π1(Y, X) = 0. Using the
relative version of the Hurewicz theorem and the induction with respect to n, we get
successively that
πn(Y, X) = Hn(Y, X) = 0.
The long exact sequence of homotopy groups for the pair (Y, X) yields that f∗ :
πn(X) → πn(Y ) is an isomorphism for all n. Applying now the Whitehead theorem
12.3 we get that f is a homotopy equivalence.
93
14. Short overview of some further methods in homotopy theory
We start this sections with two examples of computations of homotopy groups.
These computations demonstrate the fact that the possibilities of the methods we have
learnt so far are very restricted. Hence we outline some further (still very classical)
methods which enable us to prove and compute more.
14.1. Homotopy groups of Stiefel manifolds. Let n ≥ 3 and n > k ≥ 1. The
Stiefel manifold Vn,k is (n − k − 1)-connected and
πn−k(Vn,k) =



Z for k = 1,
Z for k = 1 and n − k even,
Z2 for k = 1 and n − k odd.
Proof. The statement about connectivity follows from the long exact sequence for the
fibration
Vn−1,k−1 → Vn,k → Vn,1 = Sn−1
by induction.
As for the second statement, it is sufficient to prove that
πn−2(Vn,2) =
Z for n even,
Z2 for n odd
and to use the induction in the long exact sequence for the fibration above.
We have the fibration
Sn−2
= Vn−1,1 → Vn,2
p
−→ Vn,1 = Sn−1
which corresponds to the tangent vector bundle of the sphere Sn−1
. If n is even, there
is a nonzero vector field on Sn−1
. This field is a map s : Sn−1
→ Vn,2 such that
ps = idSn−1 . Such a map is called a section and its existence ensures that the map
p∗ : πn−1(Vn,2) → πn−1(Sn−1
) is an epimorphism. Hence we get the following part of
the long exact sequence
πn−1(Vn,2)
epi
−−→ πn−1(Sn−1
)
0
−−→ πn−2(Sn−2
)
∼=
−−→ πn−2(Vn,2) → 0.
Consequently, πn−2(Vn,2) = Z.
The case n odd is more complicated. We need the fact that the Euler class of tangent
bundle of Sn−1
is twice a generator ι ∈ Hn−1
(Sn−1
). We obtain the following part of
the Gysin exact sequence for cohomology groups with integer coefficients
0 → Hn−2
(Vn,2)
0
−−→ H0
(Sn−1
)
∪2ι
−−−→ Hn−1
(Sn−1
) → Hn−1
(Vn,2) → 0.
From this sequence and the universal coefficient theorem we get that
0 = Hn−2
(Vn,2; Z) ∼= Hom(Hn−2(Vn,2), Z)
Z2
∼= Hn−1
(Vn,2) ∼= Hom(Hn−1(Vn,2), Z) ⊕ Ext(Hn−2(Vn,2), Z)
which implies that Hn−2(Vn,2; Z) ∼= Z2. The Hurewicz theorem now yields πn−1(Vn,2) ∼=
Z2.
94
14.2. Hopf fibration. Consider the Hopf fibration
S1
→ S3 η
−−→ S2
defined in 10.5. From the long exact sequence for this fibration we get
πi(S2
) ∼= πi(S3
) for i ≥ 2.
Particularly,
π3(S2
) ∼= Z
with [η] as a generator (since [id] is a generator of π3(S3
)). By the Freudenthal theorem
Z ∼= π3(S2
)
epi
−→ π4(S3
)
∼=
−→ πs
1. The methods we have learnt so far give us only that
π4(S3
) ∼= πs
1 is a factor of Z with Ση as a generator.
Exercise. Try to compute as much as possible from the long exact sequences for the
other two Hopf fibrations in 10.5.
14.3. Composition methods were developed in works of I. James and the Japanese
school of H. Toda in the 1950-ies and are described in the monograph [Toda]. They
enable us to find maps which determine the generators of homotopy groups πn+k(Sn
)
for k not very big (approximately k ≤ 20). For these purposes various types of
compositions and products are used.
Having two maps f : Si
→ Sn
and g : Sn
→ Sm
their composition gf : Si
→ Sm
determines an element [gf] ∈ πi(Sm
) which depends only on [f] and [g]. If the target
of f is different from the source of g, we can use suitable multiple suspensions to be
able to make compositions. For instance, if f : S6
→ S4
and g : S7
→ S3
we can make
composition g ◦ (Σ3
f) : S9
→ S3
. (Here Σ stands for reduced suspension.) In this way
we get a bilinear map πs
a × πs
b → πs
a+b.
More complicated tool is the Toda bracket. Consider three maps
W
f
−→ X
g
−→ Y
h
−→ Z
preserving distinquished points such that gf ∼ 0 and hg ∼ 0. Then gf can be extended
to a map F : CW → Y and hg can be extended to a map G : CX → Z. (C stands
for reduced cone.) Define f, g, h : ΣW = C+W ∪ C−W → Z as G ◦ Cf on C+W and
h ◦ F on C−W.
This definition depends on homotopies gf ∼ 0 and hg ∼ 0. So it defines a map
from πs
i × πs
j × πs
k to cosets of πs
i+j+k+1. See [Toda] and also Exercise 39 in [Hatcher],
Chapter 4.2.
The Whitehead product [ , ] : πi(X) × πj(X) → πi+j−1(X) is defined as follows:
f : Ii
→ X and g : Ij
→ X define the map f × g : Ii+j
= Ii
× Ij
→ X and we put
[f, g] = (f × g)/∂Ii+j
.
Having a map f : S2n−1
→ Sn
, n ≥ 2, we can construct a CW-complex Cf =
Sn
∪f e2n
with just one cell in the dimensions 0, n and 2n. Denote the generators of
Hn
(Cf ; Z) and H2n
(Cf ; Z) by α and β, respectively. Then the Hopf invariant of f is
the number H(f) such that
α2
= H(f)β.
95
W X Y Z
f g h
F
Cf G
Figure 14.1. Definition of Toda bracket < f, g, h >.
The Hopf invariant determines a homomorphism H : π2n−1(Sn
) → Z.
For the Hopf map η : S3
→ S2
we have Cη
∼= CP2
, consequently
H(η) = 1.
For id : S2
→ S2
we can make the Whitehead product [id, id] : S3
→ S2
and compute
(see [Hatcher], page 474) that
H([id, id]) = ±2.
Since π3(S2
) ∼= Z, we get [id, id] = ±2η. One can show (see [Hatcher], page 474 and
Corollary 4J.4) that the kernel of the suspension homomorphism π3(S2
) → π4(S3
) is
generated just by [id, id]. By the Freudental theorem this suspension homomorphism
is an epimorphism which implies that
π4(S3
) ∼= Z2.
Consequently, πs
1
∼= Z2.
Remark. J. F. Adams proved in [Adams1] that the only maps with the odd Hopf
invariant are the maps coming from the Hopf fibrations S3
→ S2
, S7
→ S4
and
S15
→ S8
.
Another important tool for composition methods is the EHP exact sequence for the
homotopy groups of Sn
, Sn+1
and S2n
:
π3n−2(Sn
)
E
−→ π3n−1(Sn+1
)
H
−→ π3n−2(S2n
)
P
−→ π3n−3(Sn
) → . . .
· · · → πi(Sn
)
E
−→ πi+1(Sn+1
)
H
−→ πi(S2n
)
P
−→ πi−1(Sn
) → . . .
Here E stands for suspension, H refers to a generalized Hopf invariant and P is defined
with connection to the Whitehead product. See [Whitehead], Chapter XII or [Hatcher],
page 474.
For n = 2 the EHP exact sequence yields
π4(S2
)
E
−→ π5(S3
)
H
−→ π4(S4
)
P
−→ π3(S2
)
E
−→ π4(S3
) → 0.
96
Since π4(S3
) ∼= Z2, π3(S2
) ∼= Z and π4(S4
) ∼= Z, we obtain that P is a multiplication
by 2 and H = 0. From the long exact sequence for the Hopf fibration (see 14.2) we
get that π4(S2
) ∼= π4(S3
) ∼= Z2 with the generator η(Ση). So π5(S3
) is either Z2 or 0.
By a different methods one can show that
π5(S3
) ∼= Z2
with the generator (Ση)(Σ2
η).
14.4. Cohomological methods have been playing an important role in homotopy
theory since they were introduced in the 1950-ies.
By the methods used in proofs in Section 12 we can construct so called EilenbergMcLane
spaces K(G, n) for any n ≥ 0 and any group G, Abelian if n ≥ 2. These spaces
are up to homotopy equivalence uniquely determined by their homotopy groups
πi(K(G, n)) =
0 for i = n,
G for i = n.
Moreover, these spaces provide the following homotopy description of reduced singular
cohomology groups
[(X, ∗), (K(G, n), ∗)]
∼=
−−→ Hn
(X; G).
To each [f] ∈ [(X, ∗), (K(G, n), ∗)] we assign
f∗
(ι) ∈ Hn
(X; G)
where ι is the generator of
Hn
(K(G, n); G) ∼= Hom(Hn(K(G, n); Z), G) ∼= Hom(G, G)
corresponding to idG.
A system of homomorphisms θX : Hn
(X; G1) → Hm
(X; G2) which is natural, i. e.
f∗
θY = θXf∗
for all f : X → Y , is called a cohomology operation. A system of cohomology
operations θj : Hn+j
→ Hm+j
is called stable if it commutes with suspensions
Σθj = θj+1Σ.
The most important stable cohomology operations for singular cohomology are the
Steenrod squares and the Steenrod powers:
Sqi
: Hn
(X; Z2) → Hn+i
(X; Z2)
Pi
p : Hn
(X; Zp) → Hn+2i(p−1)
(X; Zp) for p = 2 a prime.
For their definition and properties see [SE] or [Hatcher], Section 4.L. These operations
can be also interpreted as homotopy classes of maps between Eilenberg-McLane spaces,
for instance
Sqi
: K(Z2, n) → K(Z2, n + i).
Example A. We show how the Steenrod squares can be used to prove that some
maps are not homotopic to a trivial one. Consider the Hopf map η : S3
→ S2
. We
97
know that Cη = CP2
and H2
(CP2
; Z2) and H4
(CP2
; Z2) have generators α and α2
.
Since one of the properties of the Steenrod squares is
Sqn
x = x2
for x ∈ Hn
(X; Z2),
we get that Sq2
α = α2
= 0. Using this fact we show that [Ση] ∈ π4(S3
) is nontrivial.
For reduced cones and reduced suspensions one can prove that
CΣη = ΣCη ΣCP2
.
Then Σα : ΣCP2
→ K(Z2, 3) and Σα2
: ΣCP2
→ k(Z2; 5) represent generators in
H3
(ΣCP2
; Z2) and H5
(ΣCP2
; Z2), respectively. Now
Sq2
(Σα) = Σ(Sq2
α) = Σα2
= 0.
If Ση were homotopic to a constant map, we would have CΣη = S3
∨ S5
, and consequently,
Sq2
(Σα) = 0 since Sq2
is trivial on S3
.
Example B. We outline how to compute πn+1(Sn
) using cohomological methods.
A generator α ∈ Hn
(Sn
) induces up to homotopy a map Sn
→ K(Z, n). Further,
Hn
(K(Z, n); Z) ∼= Z with a generator ι and Hn+2
(K(Z, n); Z2) ∼= Z2 with the generator
Sq2
ρι where ρ : Hn
(X; Z) → Hn
(X; Z2) is induced by reduction mod 2. Sq2
ρι induces
up to homotopy a map
K(Z, n)
Sq2ρι
−−−→ K(Z2, n + 2).
Consider the fibration
ΩK(Z2, n + 2) → PK(Z2, n + 2) → K(Z2, n + 2)
where PX is the space of all maps p : I → X, p(1) = x0 and ΩX is the space of all
maps ω : I → X, ω(0) = ω(1) = x0. (These maps are called loops in X.) One can
show that ΩK(Z2, n + 2) has a homotopy type of K(Z2, n + 1). The pullback of the
fibration above by the map Sq2
ρι : K(Z, n) → K(Z2, n + 2) is the fibration
K(Z2, n + 1) → E
p
−→ K(Z, n).
Since Sq2
ρα = 0 in Hn+2
(Sn
; Z), one can show that the map α : Sn
→ K(Z, n) can
be lifted to a map f : Sn
→ E.
E
p

Sn
α
//
f
::vvvvvvvvvv
K(Z, n)
One can compute f∗
in cohomology (using so called long Serre exact sequence) and then
also f∗ in homology. A modified version of the homology Whitehead theorem implies
that f is an (n + 2)-equivalence. Hence f∗ : πn+1(Sn
) → πn+1(E) is an isomorphism.
Using the long exact sequence for the fibration (E, K(Z, n), p) we get
Z2
∼= πn+1(K(Z2, n + 1))
∼=
−−→ πn+1(E) ∼= πn+1(Sn
).
For more details see [MT].
98
The Steenrod operations form a beginning for the second course in algebraic topology
which should contain spectral sequences, other homology and cohomology theories,
spectra. We refer the reader to [Adams2], [Kochman], [MT], [Switzer], [Whitehead] or
to the last sections of [Hatcher].
99
References
[A1] J. F. Adams, On the non-existence of Hopf invariant one, Ann. of Math. 72 (1960), 20–104.
[A2] J. F. Adams, Stable homotopy and generalised homology, The University of Cicago Press (1974).
[Br] G. E. Bredon, Topology and geometry, Graduate Texts in Math. Springer (1997).
[Do] A. Dold, Lectures on Algebraic Topology, Springer (1972).
[Ha] A. Hatcher, Algebraic Topology, Cambridge University Press (2002).
[Ko] S. O. Kochman, Bordisms, Stable Homotopy and Adams Spectral Sequences, Fields Institutes
Monographs 7, AMS, (1996).
[MS] J. W. Milnor, J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton,
(1974).
[MT] R. E. Mosher, M. C. Tangora, Cohomology operations and applications in homotopy theory,
Harper & Row Publishers, New York, (1968).
[Sp] E. H. Spanier, Algebraic Topology, McGraw-Hill Book Company, New York, (1966).
[SE] N. E. Steenrod, D. B. A. Epstein, Cohomology Operations, Annals of Math. Sudies, 50, Princeton
Univ. Press, (1962).
[Sw] R. M. Switzer, Algebraic Topology – Homotopy and Homology, Springer, (1975).
[To] H. Toda, Composition Methods in the Homotopy Groups of Spheres, Amm. Math. Studies,
Princeton, (1962).
[Wh] G. W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Math., Springer, (1995).
100
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