топология / Tom Dieck T. Algebraic topology (EMS, 2008)

Suppose that f that induces isomorphisms j .A/ ! j .B/ and j .X/ !j .Y / for j 2 fn; n C 1g, n 1. Then the Five Lemma (11.1.4) implies that f W nC1.X; A; / ! nC1.Y; B; / is an isomorphism. With some care, this also holds for n D 0, see Problem 3.

Let f W .X; A/ ! .Y; B/ be a map of pairs such that the individual maps X ! Y and A ! B induce for each base point in A isomorphism for all n, then f W n.X; A; a/ ! n.Y; B; f .a// is bijective for each n 1 and each a 2 A. For the case n D 1 see Problem 3.

The transport functors have special properties in low dimensions.

(6.2.5) Proposition. Let v W I ! X be given. Then v# W 1.X; v.0// ! 1.X; v.1//

Proof. We first prove the claim in a universal situation and then transport it by naturality to the general case. Set D D D.2/; S D S.1/.

Let 1; 2 2 2.D _ D; S _ S/ be the elements represented by the inclusions of the summands .D; S/ ! .D _ D; S _ S/. Set D @. 2/ 2 1.S _ S/. From (6.2.3) and (6.2.5) we compute

@. 1 / D .@ 1/ D 1.@ 1/ D .@ 2/ 1.@ 1/.@ 2/ D @. 2 1 1 2/:

Since D _ D is contractible, @ is an isomorphism, hence 1 2 D 2 1 1 2.

Let now h W .D _ D; S _ S/ ! .X; A/ be a map such that hik represents xk , i.e., h . k / D xk . The computation

The actions of the fundamental group also explain the difference between

induces an injective map of the orbits of the 1.A; /-action; this map is bijective if A is path connected .n 2/.

Problems

1. Let A be path connected. Each element of 1.X; A; a/ is represented by a loop in .X; a/. The map j W 1.X; a/ ! 1.X; A; a/ induces a bijection of 1.X; A; a/ with the right (or left) cosets of 1.X; a/ modulo the image of i W 1.A; a/ ! 1.X; a/.

2. Let x 2 1.X; A; a/ be represented by v W I ! X with v.1/ 2 A and v.0/ D a. Let w W I ! X be a loop in .X; a/. The assignment .Œw ; Œv / 7!Œw v D Œw Œv defines a left action of the group 1.X; a/ on the set 1.X; A; a/. The orbits of this action are the pre-images of elements under @ W 1.X; A; a/ ! 0.A; a/. Let .F; f / W .X; A/ ! .Y; B/ be a map of pairs. Then F W 1.X; A; a/ ! 1.Y; B; f .a// is equivariant with respect to the homomorphism F W 1.X; a/ ! 1.Y; f .a//. Let Œv 2 1.X; A; a/ with v.1/ D u 2 A. The isotropy group of Œv is the image of 1.A; u/ in 1.X; a/ with respect to Œw 7!Œv w v . Find an example ˛0; ˛1 2 1.X; A; a/ such that ˛0 has trivial and ˛1 non-trivial isotropy group. It is in general impossible to define a group structure on1.X; A; a/ such that 1.X; a/ ! 1.X; A; a/ becomes a homomorphism.

3. Although there is only a restricted algebraic structure at the beginning of the exact sequence we still have a Five Lemma type result. Let f W .X; A/ ! .Y; B/ be a map of pairs. If f W 0.A/ ! 0.B/ and f W 1.X; a/ ! 1.Y; f .a// are surjective and f W 0.X/ !

0.Y / is injective, then f W 1.X; A; a/ ! 1.Y; B; f .a// is surjective. Suppose that for each c 2 A the maps f W 1.X; c/ ! 1.Y; f .c// and f W 0.A/ ! 0.B/ are injective and f W 1.A; c/ ! 1.B; f .c// is surjective, then f W 1.X; A; a/ ! 1.Y; B; f .a// is injective for each a 2 A.

4.Let .X; A/ be a pair such that X is contractible. Then @ W qC1.X; A; a/ ! q .A; a/ is for each q 0 and each a 2 A a bijection.

5.Let A X be an h-equivalence. Then n.X; A; a/ D 0 for n 1 and a 2 A.

6.Let X carry the structure of an h-monoid. Then 1.X/ is abelian and the action of the

fundamental group on n.X; / is trivial.

application of the functor Π; X 0. If we use the model Dn=Sn 1 for the n-sphere, then an explicit map n is x 7!.2x; / for 2kxk 1 and x 7!. ; 2kxk 1/ for 2kxk 1.

6.3 Serre Fibrations

The notion of a Serre fibration is adapted to the investigation of homotopy groups, only the homotopy lifting property for cubes is used.

(6.3.1) Theorem. Let p W E ! B be a Serre fibration. For B0 B set E0 D p 1B0. Choose base points 2 B0 and 2 E0 with p. / D . Then p induces for n 1 a bijection p W n.E; E0; / ! n.B; B0; /.

We then have H.@I / E0, and therefore H represents a pre-image of x under p .

sequence of the pair .E; F; / and obtain as a corollary to (6.3.1) the exact sequence of a Serre fibration:

(6.3.2)Theorem. For a Serre fibration p W E ! B with inclusion i W F D p 1.b/ E and x 2 F the sequence

The new map @ has the following description: Let f W .I n; @I n/ ! .B; b/ be given. View f as I n 1 I ! B. Lift to W I n ! E, constant on J n 1. Then @Œf is represented by jI n 1 1. The very end of the sequence requires a little extra argument. For additional algebraic structure at the beginning of the sequence see the discussion of the special case in (3.2.7).

(6.3.3) Theorem. Let p W E ! B be a continuous map and U a set of subsets such that the interiors cover B. Assume that for U 2 U the map pU W p 1.U / ! U induced by p is a Serre fibration. Then p is a Serre fibration.

the Lebesgue lemma (2.6.4). Let V k I n denote the union of the k-dimensional faces of the subdivision of I n.

We have to solve a lifting problem for the space I n with initial condition a. We begin by extending a over I n Œ0; ı to a lifting of h. We solve the lifting problems

for k D 0; : : : ; n with V 1 D ; and H. 1/ D a by induction over k. Let W be a k-dimensional cube and @W the union of its .k 1/-dimensional faces. We can solve the lifting problems

by a map HW , since pU is a Serre fibration; here U 2 U was chosen such that

(6.3.4) Example. Since a product projection is a fibration we obtain from (6.3.3): A locally trivial map is a Serre fibration. Þ

is a Serre fibration with fibre n.F; e/.

6.4 The Excision Theorem

A basic result about homotopy groups is the excision theorem of Blakers and Massey [22].

(6.4.1) Theorem (Blakers–Massey). Let Y be the union of open subspaces Y1 and Y2 with non-empty intersection Y0 D Y1 \ Y2. Suppose that

for each base point 2 Y0. Then the excision map, induced by the inclusion,

W n.Y2; Y0; / ! n.Y; Y1; /

is surjective for 1 n p C q 2 and bijective for 1 n < p C q 2 ( for each choice of the base point 2 Y0/. In the case that p D 1, there is no condition on

i .Y1; Y0; ).

We defer the proof of this theorem for a while and begin with some applications and examples. We state a special case which has a somewhat simpler proof and already interesting applications. It is also a special case of (6.7.9).

(6.4.2) Proposition. Let Y be the union of open subspaces Y1 and Y2 with non-

We apply the excision theorem (6.4.1) to the homotopy group of spheres. We use the following subspaces of Sn, n 0,

D˙n D f.x1; : : : ; xnC1/ 2 Sn j ˙xnC1 0g H˙n D fx 2 Sn j x 6D enC1g:

134 Chapter 6. Homotopy Groups

For n 0 we have a diagram with the isomorphisms (6.4.3)

The morphism is induced by the inclusion and E is defined so as to make the diagram commutative. Note that the inductive proof of (1) in the next theorem only uses (6.4.2).

(6.4.4) Theorem. .1/ i .Sn/ D 0 for i < n.

.2/ The homomorphism is an isomorphism for i 2n 2 and an epimorphism for i D 2n 1. A similar statement holds for E.

Proof. Let N.n/ be the statement (1) and E.n/ the statement (2). Obviously N.1/ holds. Assume N.n/ holds. We then deduce E.n/. We apply the excision theorem

for p D q D n C 1 and see that is surjective for i C 1 2n and bijective for

(6.4.5) Proposition. The homomorphism i .DnC1; Sn; / ! i .DnC1=Sn; / induced by the quotient map is an isomorphism for i 2n 1 and an epimorphism for i D 2n.

Proof. Consider the commutative diagram

The map (1) is induced by a homeomorphism and the map (2) by a homotopy equivalence, hence both are isomorphisms. Now apply (6.4.4).

The homomorphism E is essentially the suspension homomorphism. In order to see this, let us work with (6.1.4). The suspension homomorphism † is the composition

with the quotient map q W D.n C 1/ ! D.n C 1/=S.n/ D S.n C 1/.

The next result is the famous suspension theorem of Freudenthal ([66]).

(6.4.6) Theorem. The suspension † W i .S.n// ! iC1.S.n C 1// is an isomorphism for i 2n 2 and an epimorphism for i D 2n 1.

Proof. We have to show that q W iC1.CX; X/ ! iC1.CX=X/ is for X D S.n/ an isomorphism (epimorphism) in the appropriate range. This follows from (6.4.5);

(6.4.7) Theorem. n.S.n// Š Z and † W n.S.n// ! nC1.S.n C 1// is an isomorphism (n 1). The group n.S.n// is generated by the identity of S.n/.

† W 1.S.1// ! 2.S.2//; this is an isomorphism, since both groups are isomorphic to Z. For n 2, (6.4.6) gives directly an isomorphism † . We know that1.S.1// Š Z is generated by the identity, and † respects the identity.

(6.4.8) Example. We continue the discussion of the Hopf fibrations (6.3.6). The

sion S2nC1 ! S2nC3, z 7!.z; 0/ induces an embedding CP n CP nC1. We

compare the corresponding Hopf fibrations and their exact sequences and conclude

2.CP n/ Š 2.CP nC1/. Let CP 1 D Sn 1 CP n be the colimit. The canonical inclusion CP n CP 1 induces i .CP n/ Š i .CP 1/ for i 2n. A proof uses

the fact that a compact subset of CP 1 is contained in some finite CP N . Therefore CP 1 is a space with a single non-trivial homotopy group 2.CP 1/ Š Z.

Note also the special case 3.S2/ Š 3.S3/ Š Z.

We have similar results for real projective spaces. The twofold coverings

Z=2 ! Sn ! RP n are use to show that 1.RP 2/ Š 1.RP 3/ Š Š1.RP 1/ Š Z=2, induced by the inclusions, i .RP n/ Š i .RP nC1/ for i < n

6.5 The Degree

Let d W n.S.n// ! Z be the isomorphism which sends Œid to 1. If f W S.n/ ! S.n/ is a pointed map, then f W n.S.n// ! n.S.n// is the multiplication by the integer d.f / D d.f Œid / D d.Œf /. Since the map ŒS.n/; S.n/ 0 ! ŒS.n/; S.n/ which forgets about the base point is bijective (see (6.2.8)), we can transport d to a bijection d W ŒS.n/; S.n/ ! Z. The functoriality f g D .fg/ shows

136 Chapter 6. Homotopy Groups

d.fg/ D d.f /d.g/; therefore d.h/ D ˙1 if h is a homeomorphism. The suspension sends Œf to Œf ^ id ; hence d.f / D d.f ^ id/.

(6.5.1) Proposition. Given pointed maps f W S.m/ ! S.m/ and g W S.n/ ! S.n/. Then d.f ^ g/ D d.f /d.g/.

Proof. We use the factorization f ^ g D .f ^ id/.id ^g/. The map f ^ id is a suspension of f , and suspension does not change the degree. Let W S.m/^S.n/ !

is independent of the choice of kn. We use this bijection to transport d to a bijection d W ŒSn; Sn ! Z. If d.Œf / D k we call k the degree d.f / of f . We still have

the properties d.f /d.g/ D d.fg/, d.id/ D 1, d.h/ D ˙1 for a homeomorphism h. By a similar procedure we define the degree d.f / for any self-map f of a space S which is homeomorphic to S.n/.

Matrix multiplication lA W Rn ! Rn, x 7!Ax induces for each A 2 GLn.R/ a pointed map LA W S.n/ ! S.n/. For the notation see (6.1.4).

(6.5.2) Proposition. The degree of LA is the sign of the determinant det.A/.

Proof. Let w W I ! GLn.R/, t 7!A.t/ be a path. Then .x; t/ 7!LA.t/x is a homotopy. Hence d.LA/ only depends on the path component of A in GLn.R/.

The group GLn.R/ has two path components, distinguished by the sign of the determinant. Thus it suffices to show that for some A with det.A/ D 1 we have d.LA/ D 1. By the preceding discussion and (6.1.4) we see that .x1; : : : ; xn/ 7!

. x1; x2; : : : ; xn/ has degree 1.

The stereographic projection (6.1.4) now shows that the map Sn ! Sn which changes the sign of the first coordinate has degree 1.

(6.5.3) Proposition. Let A 2 O.n C 1/. Then A W Sn ! Sn, x 7!Ax has degree det.A/.

Proof. Again it suffices to verify this for appropriate elements in the two path

independent vector fields see [3].