INTRODUCTION TO ALGEBRAIC TOPOLOGY
MARTIN ČADEK
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 dln its boundary. For n = 0 we take J° to be one point and dl° to be empty. Consider a space X with a basepoint x0. Maps (In,dln) —> (X,x0) are the same as the maps of the quotient (Sn = In/dln,s0 = dln/dln) —> (X,x0). We define the n-th homotopy group of the space X with the basepoint xq as
nn(X,x0) = i(Sn,s0),(X,x0)] = i(In,dIn),(X,x0)].
7Tq(X, Xq) is the set of path connected components of X with the component containing xq as a distinguished element. For n > 1 we can introduce a sum operation on irn(X, xq)
f(2tut2,...,tn) *iG[0,§], (2£x - l,£2,...,£n) £iG[|,l].
This operation is well defined on homotopy classes. It is easy to show that irn(X,xq) is a group with identity element represented by the constant map to xq and with the inverse represented by
— fitiih, ■ ■ ■,tn) = /(I — ti,t2,... ,tn).
For n > 2 the groups irn(X,x0) are commutative. The proof is indicated by the following pictures.
(/ + p)(íl,Í2,...,*n)
	
	9
	f
9	
f
Figure 10.1. / + g ~ g + /
In the interpretation of irn(X, x0) as [(Sn, s0), (X, x0)], the sum / + g is the composition
Sn ^ Sny Sn
where c collapses the equator Sn~1 of Sn to a point s0 G Sn~1 C Sn.
l
2
Any map F : (X,x0) —y (Y,y0) induces the homomorphism : irn(X,x0) —y 7rn(y,i/q) by composition
F.([f\) = [Ff]-
Hence irn 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.
10.2. Relative homotopy groups. Consider Jn_1 as a face of In with the last coordinate tn = 0. Denote Jn_1 the closure of dln — In~x. Let (X, A) be a pair with basepoint x0 g A. For n > 1 we define the n-th relative homotopy group of the pair (X, A) as
nn(X,A,x0) = [(Dn,Sn-1,s0),(X,A,x0)] = [(In,dln, Jn_1), (X,A,x0)].
A sum operation on irn(X, A, xq) is defined by the same formula as for irn(X, xq) 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 irn(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 7rn(X, A,xq) look like. We say that two maps f,g : (Dn,Sn~1,sq) —> (X,A,xq) are homotopic rel S"1^1 if there is a homotopy h between / and g such that h(x,t) = f(x) = g{x) for all x g S"1^1 and all t g /.
Proposition. A map f : (Dn, S^1, s0) —> (X,A,x0) represents zero in irn(X, A,x0) iff' it is homotopic rel Sn~1 to a map with image in A.
Proof. Suppose that / ~ g rel S^1 and g{Dn) C A. Then g = g o id^n is homotopic to the constant map g o const into xq g A. Hence [/] = [g] = 0.
Let / be homotopic to the constant map via homotopy h : Dn x I —y X. Have a look at the picture and consider the subset
C = {(x,t) EDn x J; 2||x|| < 2 - t}
of Dn x I simultaneously with a vertical retraction r : Dn x I —y C and a horisontal homeomorphism q : C —y Dn x J.
The maps can be defined in the following way:
,    \     ((x,t) for 2||:r|| < 2 -1,
r(x t) = < ii  ii — '
'        \(x,2(l - \\x\\)    for 21|^|| > 2 -t
and
q(x,t) = yj—j;x,t
Now H = h o q o r : Dn x J —>■ X is a homotopy between iJ(x, 0) = /i(rr, 0) = f(x) and iJ(x, 1) = g{x) where
g(Dn) = H(Dn x I) = h[Dn x {1} U Sn~l x J) C A
and iJ is a homotopy rel Sn~x. □
3
Dn x I C Dn x I
figure 10.2. Retraction r and homeomorphism q
A map F : (X, A, xq) —> (Y, B, yo) induces again the homomorphism : irn(X, A, xq) —> 7Tn(Y,B,yQ). Since irn(X,xq,xq) = irn(X,xq) the functor irn on Top^ can be extended to a functor from Top^ 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 homo-topy 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
^n(A, Xq) ^> 7Tn(X, Xq) ^> 7Tn(X, A, Xq) A 7Tn_i(A, Xq) ->• . . .
where i : A X, j : (X,xq) (X, A) are inclusions and S comes from restriction, is exact.
More generally, any triple B c A c X induces the long exact sequence
•••->• nn(A,B,x0) ^> nn(X,B,x0) ^> nn(X, A,x0) -4 7rn_i(A,B,x0) ->-...
Proof. We will prove only the version for the pair (X, A).   S is defined on [/] g
nn(X, A,x0) by
s[f\ = [f/r1-1].
Exactness in irn(X,xq). According to the previous proposition = 0, hence Im«* c Kerj*. Let [/] g Kerj* for / : (In,dln) —> (X,xq). Using again the previous proposition / ~ g rel dln where g : In —> A. Hence [/] = i*[g].
Exactness in irn(X, A, x0). 5j* = 0, hence Imj* c KerS. Let [/] g KerS, i. e. f(In, dln, Jn_1) (X,A,x0) and f/I"-1 ~ const. Then according to HEP there is f\ : (In,dIn,Jn) —> (X,Xq,Xq) homotopic to /.  Therefore [f\] g 7rn(X, x0) and
[f]=Ufil
4
Exactness in irn(A, x0). Let [F] G 7rn+1(X, A, x0). Then i o F/Jn : In —> X is a map homotopic to the constant map to rro through the homotopy F. (Draw a picture.)
Let / : (In,dln) —?■ (A,x0) and / ~ 0 through the homotopy F : In x I —?■ X such that F(x,0) = f(x) G A, F/Jn = x0. Hence [F] G nn+1(X, A, x0) and S[F] = [/]. □
Remark. The boundary operator for a triple (X, A, B) is the composition
7rn(X,A) ^nn(A) ^nn^(A,B).
10.4. Changing basepoints. Let X be a space and 7 : J —>■ X a path connecting points rco and rri. This path associates to / : (In,dln) —> (X,xi) a map 7 • / : (In,dln) —> (X,xq) by shrinking the domain of / to a smaller concentric cube in In and inserting the path 7 on each radial segment in the shell between dln and the smaller cube.
x0 x0
Figure 10.3. The action of 7 on /
It is not difficult to prove that this assigment has the following properties:
(1) 7 • (/ + 9) ~ 7 • / + 7 -9 for f,g : (/", dln) -> (X, m),
(2) (7 + k) ■ f ~ 7 • (k • /) for / : (In, dln) -+ (X, x2), 7(0) = rr0, 7(1) = *i = «(0),
(3) If 7i, 72 : / —> X are homotopic rel dl = {0,1}, then 7l • / ~ 72 • /. Hence, every path 7 defines an isomorphism
7:7rn(X,7(l))^7rn(X,7(0)).
Particulary, we have a left action of the group 7Ti(X,Xq) on irn(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 x I -> E
Xx {0} Li Ax I-=p E
X x I -*■ B
5
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. // 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 — l)-skeleton to A;-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,dDk = Sk~1). The HLP for this pair follows from the fact that the pair {Dk x I, Dk x {0} U Sk~1 x J) is homeomorphic to the pair {Dk x I,Dk x {0}), see the picture below, and the fact that p has homotopy lifting property with respect to the pair (Dk, 0).
Figure 10.4. Homeomorphism (Dn x I,Dn x {0} U Sn x J)     (Dn x I,Dn x {0})
□
Proposition. Every fibre bundle (E,B,p) is a fibration.
Proof. For the definition of a fibre bundle see 8.1. Let Ua be an open covering of B with trivializations ha : p~1(Ua) —^ Ua x F. We would like to define a lift of a homotopy G : Ik x I —^ B. (We have replaced Dk by Ik.) The compactness of Ik x I implies the existence of a division
0 = to < t\ < ■ ■ ■ < tm = 1,    Ij = [tj-i,tj],
such that G(Ij± x • • • x Ijk+1) lies in some Ua. Now we make a lift H : Ik x I —^ E of G, first on (Ii)k+1 and then we add successively the other small cubes. We need retractions r of cubes C x Ijk+1 = Yli=i hi ^° a suitable part of the boundary C x {0} U A x JJfe+1 where H is already defined. A is a CW-subcomplex of the cube C and we are in the following situation
C x {0} Li Ax I UaxF
H
Pi
Cxi-- Ua
g a
Now, we can define
H(x,t) = (G(x,t),p2 ogor)(x,t) where p2 : Ua x F —> F is a projection. □
6
Example. Here you are several examples of fibre bundles.
(f) The projection p : Sn —> IRPn determines a fibre bundle with the fibre S°.
(2) The projection p : S2n+1 —> IRCn determines a fibre bundle with the fibre S1.
(3) The special case is so called Hopf fibration
S1 -> S3 -> CP1 = s2.
(4) Similarly, as complex projective space we can define quaternionic projective space HPn. The definition determines the fibre bundle
S3     S4n+S MFn.
(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 > I > 1. Then the projection
P ■ Vn)k ->• Vnj,   p(v!,v2, ...,vk) = (v±,v2, ...,vi) determines a fibre bundle with the fibre Vn-i^-i-
(9) Natural projection p : Vn^ —> Gn^ is a fibre bundle with the fibre 0(k).
10.6. Long exact sequence of a fibration. Consider a fibration p : E —>• B. Take a basepoint b0 g B, put F = p_1(&o) and choose x0 g F.
Lemma. For all n > 1
p* : nn(E, F, x0)     nn(B, b0)
is an isomorphism.
Proof. First, we show that p* is an epimorphism. Consider / : (In,dln) —> (B,b0). Let k : Jn~x —> E be the constant map into x$. Since p is a fibration the commutative diagram
jn-1 = jn-1 x |X| y Qjn-1 x j £
r-1 X I-*■ B
/
can be completed by g : (In,dln, Jn_1) —> (E,F,x0). Hence p*[g] = [/].
Now we prove that p* is a monomorphism. Consider / : (Jn, dln, Jn_1) —> (E, F, x0) such that        = 0. Then there is a homotopy G : (Jn x J, <9Jn x J) —y (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:
.r-1 x i u r x {0} u in x {1} E
H
r x i
G
B
by H : (In x I,dln x I,Jn 1 x I) -> (E,B,x0) which is a homotopy between / and the constant map k. □
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/ = /_1(&o) for f:(A,a0)^(B,b0).
Theorem. If p : E —>• B be a fibration with a fibre F = P-i(b0), x0 £ F and B is path connected, then the sequence
----> 7ln(F,X0) ^> 7ln(E,X0) ^> 7ln(B,b0) A 7Tn_i(F,X0) -> ...
is exact.
Proof. Substitute the isomorphism p* : irn(E, F,x0) —> 7rn(5,60) into the exact sequence for the pair (E,F). In this way we get the required exact sequence ending with
•••->• 7ro(F,x0) ->• n0(E,x0). We can prolong it by one term to the right. The exactness in ttq(E,Xq) 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 S : 7rn(5,60) —> ^n-i{Eixo) is given by
5[f] = [g/T-1]
where g is the lift in the diagram
jn-l x°
jn
E
B
Some applications of this long exact sequence to computations of homotopy groups will be given in Section 14.
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