INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ˇCADEK
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
1
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1
0
f−1
(pα)
f−1
(q)
ϕ
π
Ii−1
Figure 13.1. The graph of ϕ
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.)
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).
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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
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
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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.
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).
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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
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.
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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
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.
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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.
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.
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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.
CZ.1.07/2.2.00/28.0041
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