INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ČADEK
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, intX and dX, respectively. The letter I will stand for the interval [0,1]. IRn and Cn will denote the vector spaces of n-tuples of real and complex numbers, respectively, with the standard norm ||rr|| = Y^i=i \xi\2■ The sets
Dn = {xe Rn; \\x\\ < 1},
Sn = {xe 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 / : A —>• B and g : B —>• C can be composed in a morphism g o / : A —>• C and for every object B there is a morphism idß : B —> B such that ids of = f and g o ids = 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 / : (X, *) —> (Y, *) such that /(*) = * 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 / : (X, A) —> (Y, B) means that / : X —> Y is a continuous map which preserves subspaces, i. e. f{A) C B. The category Top2 consists of pairs of topological spaces as objects and continuous maps / : (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 / ~ g, if there is a map /i:Ix/->7 such that h(x,0) = f{x) and h(x, 1) = g{x). This map is called homotopy between / and g. The relation ~ is an equivalence. Homotopies in categories Top^, Top2 or Top2 have to preserve distinquished points, i. e. h(*,i) = *, subsets or both subsets and distinquished points, respectively.
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Spaces X and Y are called homotopy equivalent if there are maps / : X —>• Y and g : Y —> X such that / o g ~ idy and jo/~ idx. We also say that the spaces X and y have the same homotopy type. The maps / and g are called homotopy equivalences.
A space is called contractible if it is homotopy equivalent to a point.
Example. Sn and IRn+1 — {0} are homotopy equivalent. As homotopy equivalences take the inclusion / : Sn     Rn+1 - {0} and # : Rn+1 - {0}     Sn, g(x) = x/\\x\\.
1.4. Retracts and deformation retracts. Let « : A H> X be an inclusion. We say
that A is a retract of X if there is a map r : X —>• A such that r o i = id a- The map r is called a retraction.
We say that A is a deformation retract of X (sometimes also strong deformation retract) if i o r : X —> A —y X is homotopic to the identity on X relative to A, i.e. there is a homotopy h : X x I —>• X such that /?.(—, 0) = idx, h(—, 1) = i or and h(i(—),t) = idA for all t £ I. The map /i 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 C X/~ is open iff is open in X.
Exercise A. The map / : (X/~) —> Y is continuous iff the composition / op : 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 Xj~ where rr ~ y iff rr = 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 = 0 we put X/0 = X U {*}.
Exercise B. Prove that Dn/Sn~1 is homeomorphic to Sn. For it consider / : Dn —> Sn
f(xt, x2,.. . , xn) = (2a/1 - ||:r||2:r, 2||x||2 - 1).
Disjoint union of spaces X and Y will be denoted X UY. Open sets are unions of open sets in X and in Y. Let A be a subspace of X and let / : A —> Y be a map. Then IU/F is the space (X U7)/~ where the equivalence is generated by relations a ~ f(a).
The mapping cylinder of a map / : X —>■ F is the space
Mf =X x JU/xiF
which arises from X x J and F after identification of points (rr, 1) G X x I and /(*) G Y.
X
f			
	'C		V
f{X)
Mt
figure 1.1. Mapping cylinder
Exercise C. We have two inclusions ix '■ X = X x {0} Mf and iy : Y Mf and a retraction r : Mf —> Y. How is r defined?
X
Y
Mf
Y
iY
Prove that
(1) Y is a deformation retract of Mf,
(2) ixor = f,
(3) «yo/~ ix.
The mapping cone of a mapping / : X —> Y is the space
C> = Mf/(X x {0}). A special case of a mapping cone is the cone of a space X
CX = Xx J/(Xx{0}) = Cidx. The suspension of a space X is the space
SX = CX/(X x {1}). Exercise D. Show that SSn = Sn+1. For it consider the map / : Sn x I —> Sn+1
f(x, t) = (y/1- (2t- l)2x, 2t - 1). The join of spaces X and Y is the space
where ~ is the equivalence generated by (x, y±,0) ~ (x, y2, 0) and (xi,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, S°*X, Sn*Sm.
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1.6. Basic constructions in Top^ and Top2. Let X be a space with a base point xq. The reduced suspension of X is the space
EX = SX/({x0} x J)
with base point determined by xq x J. In the next section in ?? we will show that EX is homotopy equivalent to SX. The space
(X,x0) v (y,y0) = X x {y0} U {x0} x y with distinguished point (xQ,y0) is called the wedge of X and y and usually denoted only as X V y.
The smash product of spaces (X,xq) and (Y,yo) is the space
x a y = x x y/(x x {y0} u {x0} x y) = x x y/x v y.
Analogously, the smash product of pairs (X, A) and (Y, B) is the pair
(X x Y,A x y UX x B). Exercise A. Show that Sm a Sn = Sn+m. One way how to do it is to prove that
XIA a Y/B = X x Y/A xYUX xB.
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 / : X —y Y and any homotopy h : A x I —y Y such that h(a, 0) = f(a) for a £ A, and
fUh:Xx{0}UAxI^Y
is continuous, can be extended to a homotopy H : Xxl ^Y such that 0) = f(x) and H(a,t) = h(a,t) for all rr G X, a G A and t £ I, i.e. iJ is an arrow making the diagram
X x {0}UAx/^ly X x /
commutative. If the pair (X, A) satisfies HEP, we call the inclusion A4la cofibra-tion.
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figure 1.2. Homotopy extension property Theorem. A pair (X, A) has HEP if and only if X x {0} U Ax I is a retract of X x I.
X x {0}
Figure 1.3. Retraction XxI->Xx{0}UAxI
Exercise A. Using this Theorem show that the pair (Dn,Sn~1) satisfies HEP. Many other examples will be given in the next section.
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\
\ \
/
/
/
/ t—
\ \
r
X
>>    Z = r(Z)
/
/
/
\
if
r(x)
r(y)
Figure 1.4. Retraction D1 x I     D1 x {0} U S° x I
Proof of Theorem. Let (X, A) has HEP. Put Y = X x {0} U A x I and consider / U h : X x {0} U A x J —>• X x {0} U A x I to be an identity. Its extension H : X x J —>• X x {0} U A x J is a retraction.
Let r : X x I —y X x {0} U A x J be a retraction. Given a map / and a homotopy h as in the definition which together determine a continuous map F = (f U h) : X x {0} U A x I —> Y, then H = F o r is an extension of / U h. □
Exercise B. Let a pair (X, A) satisfy HEP and consider a map g : A —> Y. Prove that (X Ug 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 G IR; n = 0,1, 2,... } U {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 / : X —y Y. Show that the inclusion ix '■ X M/ is a cofibration. In particular, the map / : X —>■ F can be factored into the composition roix of the cofibration ix and the homotopy equivalence r. (See the exercise after the definition of the mapping cylinder.)
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