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

conceptual reformulation (6.9.3). The reformulation is more satisfactory, because it is “symmetric” in Y1; Y2. In (6.4.1) we have a second conclusion with the roles of Y1 and Y2 interchanged.

We begin with a technical lemma used in the proof. A cube in Rn, n 1 will be a subset of the form

W D W .a; ı; L/ D fx 2 Rn j ai xi ai C ı for i 2 L; ai D xi for i … Lg

for a D .a1; : : : ; an/ 2 Rn, ı > 0, L f1; : : : ; ng. (L can be empty.) We set dim W D jLj. A face of W is a subset of the form

of all faces of W which are different from W . We use the following subsets of

W D W .a; ı; L/:

Here 1 p n. For p > dim W we let Kp.W / and Gp.W / be the empty set.

(6.9.1) Lemma. Let f W W ! Y and A Y be given. Suppose that for p dim W the inclusions

f 1.A/ \ W 0 Kp.W 0/ for all W 0 @W

hold. Then there exists a map g which is homotopic to f relative to @W such that

x to P .y/ affinely to the segment from x to Q.y/. Then h is homotopic relative to @I n to the identity. We set g D f h. Let z 2 I n and g.z/ 2 A. If zi < 12 for all i,

The next theorem is the basic technical result. In it we deform a map I n ! Y into a kind of normal form. We call it the preparation theorem. Let Y be the union

150 Chapter 6. Homotopy Groups

of open subspaces Y1; Y2 with non-empty intersection Y0. Let f W I n ! Y be given. By the Lebesgue lemma (2.6.4) there exists a subdivision of I n into cubes W such that either f .W / Y1 or f .W / Y2 for each cube. In this situation we claim:

(6.9.2) Theorem. Suppose .Y1; Y0/ is p-connected and .Y2; Y0/ is q-connected (p; q 0). Then there exists a homotopy ft of f with the following properties:

(1)If f .W / Yj , then ft .W / Yj .

(2)If f .W / Y0, then ft is constant on W .

(3)If f .W / Y1, then f1 1.Y1 X Y0/ \ W KpC1.W /.

(4)If f .W / Y2, then f1 1.Y2 X Y0/ \ W GqC1.W /.

Here W is any cube of the subdivision.

Proof. Let C k be the union of the cubes W with dim W k. We construct the homotopy inductively over C k I .

Let dim W D 0. If f .W / Y0 we use condition (2). If f .W / Y1; f .W / 6 Y2, there exists a path in Y1 from f .W / to a point in Y0, since .Y1; Y0/ is 0- connected. We use this path as our homotopy on W . Then (1) and (3) hold. Similarly if f .W / Y2; f .W / 6 Y1. Thus we have found a suitable homotopy on C 0. We extend this homotopy to the higher dimensional cubes by induction over the dimension; we use that @W W is a cofibration, and we take care of (1) and (2).

Suppose we have changed f by a homotopy such that (1) and (2) hold and (3),

(4) for cubes of dimension less than k. Call this map again f . Let dim W D k. If f .W / Y0, we can use (2) for our homotopy. Let f .W / Y1; f .W / 6 Y2. If dim W p, there exists a homotopy ftW W W ! Y1 relative to @W of f jW with f1W .W / Y0, since .Y1; Y0/ is p-connected. If dim W > p we use (6.9.1) in order to find a suitable homotopy of f jW . We treat the case f .W / Y2; f .W / 6 Y1 in a similar manner. Again we extend the homotopy to the higher dimensional cubes. This finishes the induction step.

Let us denote by F .Y1; Y; Y2/ the path space fw 2 Y I j w.0/ 2 Y1; w.1/ 2 Y2g. We have the subspace F .Y1; Y1; Y0/.

(6.9.3) Theorem. Under the hypothesis of the previous theorem the inclusion

F .Y1; Y1; Y0/ F .Y1; Y; Y2/ is .p C q 1/-connected.

Proof. Let a map ' W .In; @In/ ! .F .Y1; Y; Y2/; F .Y1; Y1; Y0// be given where n p C q 1. We have to deform this map of pairs into the subspace. By adjunction, a map of this type corresponds to a map ˆ W I n I ! Y with the following properties:

(1)ˆ.x; 0/ 2 Y1 for x 2 I n,

(2)ˆ.X; 1/ 2 Y2 for x 2 I n,

(3) ˆ.y; t/ 2 Y1 for y 2 @I n and t 2 I .

Let us call maps of this type admissible. The claim of the theorem is equivalent to the statement, that ˆ can be deformed as an admissible map into a map with image in Y1. We apply the preparation theorem to ˆ and obtain a certain map ‰.

The deformation in (6.9.2) stays inside admissible maps. Consider the projectionW I n I ! I n. We claim that the images of ‰ 1.Y X Y1/ and ˆ 1.Y X Y2/ under are disjoint. Let y 2 ‰ 1.Y X Y2/, y D .z/ and z 2 ‰ 1.Y X Y2/\ W for a cube W . Then z 2 KpC1.W / and hence y has at least p small coordinates. In

a similar manner we conclude from y 2 ‰ 1.Y X Y1/ that y has at least q large coordinates. In the case that n < p C q the point y cannot have p small and q large coordinates.

The set ‰ 1.Y X Y1/ is disjoint to @I n, since ‰.@I n/ I / A. There exists

(6.9.4) Theorem. Under the hypothesis of (6.9.2) the inclusion induces an isomorphism j .Y1; Y0/ ! j .Y; Y2/ for j < p C q and an epimorphism for j D p C q.

Proof. We have the path fibration F .Y; Y; Y2/ ! Y , w 7!w.0/. The pullback along Y1 Y yields the fibration F .Y1; Y; Y2/ ! Y1, w 7!w.0/. The fibre over

is F . ; Y; Y2/. We obtain a commutative diagram of fibrations:

The inclusion ˛ is .p C q 1/-connected (see (6.9.3)). Hence ˇ has the same connectivity (see (6.7.8)), i.e., the inclusion .Y1; Y0/ .Y; Y2/,

induces an isomorphism for n < p C q 1 and an epimorphism for n D p C q 1.

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Problems

1. The hypothesis of (6.4.1) is a little different from the hypothesis of (6.9.4), since we did not assume in (6.4.1) that .Y1; Y0/ and .Y2; Y0/ are 0-connected. Let Y 0 be the subset of points that can be connected by a path to Y0. Show that Y 0 has the open cover Y10 ; Y20 and the inclusion induces isomorphisms .Y10 ; Y0/ Š .Y1; Y0/ and .Y; Y2/ Š .Y 0; Y20 /. This reduces (6.4.1) to (6.9.4).

6.10 Further Applications of Excision

The excision theorem is a fundamental result in homotopy theory. For its applications it is useful to verify that it holds under different hypotheses. In the next proposition we show and use that Y is the homotopy pushout.

(6.10.1) Proposition. Let a pushout diagram be given with a cofibration j ,

f

AB

jJ

Suppose i .X; A; a/ D 0 for 0 < i < p and each a 2 A, and i .f; a/ D 0 for

0 < i < q and each a 2 A. Then the map .F; f / W n.X; A; a/ ! n.Y; B; f .a// is surjective for 1 n p C q 2 and bijective for 1 n < p C q 2.

Proof. We modify the spaces up to h-equivalence such that (6.4.1) can be applied. Let Z.f / D B[f A Œ0; 1 D BCA Œ0; 1 =f .a/ .a; 0/ be the mapping cylinder of f with inclusion k W A ! Z.f /, a 7!.a; 1/ and projection p W Z.f / ! B a homotopy equivalence. We form the pushout diagrams

with pk D f and PK D F . Then P is a homotopy equivalence by (5.1.10) and .P; p/ induces an isomorphism of homotopy groups. Therefore it suffices to analyze .K; k/ . The space Z can be constructed as

Z D B [f A Œ0; 1 [ X D Z.f / C X=.a; 1/ a:

The map .K; k/ is the composition of

.X; A; a/ ! .A 0; 1 [ X; A 0; 1 ; .a; 1//; x 7!.x; 1/

with the inclusion into .Z; Z.f /; .a; 1//. The first map induces an isomorphism of homotopy groups, by homotopy equivalence. In order to exhibit n. / as an isomorphism, we can pass to the base point .a; 1=2/, by naturality of transport. With this base point we have a commutative diagram

The vertical maps are isomorphism, by homotopy invariance. We apply (6.4.1) to the bottom map. Note that i .A 0; 1 [ X; A 0; 1 / Š i .X; A/ and

(6.10.2) Theorem (Quotient Theorem). Let A X be a cofibration. Let further p W .X; A/ ! .X=A; / be the map which collapses A to a point. Suppose that for each base point a 2 A,

i .CA; A; a/ D 0 for 0 < i < m; i .X; A; a/ D 0 for 0 < i < n:

Then p W i .X; A; a/ ! i .X=A; / is bijective for 0 < i < m C n 2 and surjective for i D n C m 2.

Proof. By pushout excision, i .X; A/ ! i .X [ CA; CA/ is bijective (surjective) in the indicated range. Note that @ W i .CA; A; a/ Š i 1.A; a/, so that the first hypothesis is a property of A. The inclusion CA X[CA is an induced cofibration. Since CA is contractible, the projection p W X [ CA ! X [ CA=CA Š X=A is a

0 i m 1 and i .X/ D 0 for 0 i m 2. Then i .X; A/ ! i .X=A/

154 Chapter 6. Homotopy Groups

is an isomorphism for 0 < i 2m 1. We use this isomorphism in the exact sequence of the pair .X; A/ and obtain an exact sequence

2m 1.A/ ! 2m 1.X/ ! 2m 1.X=A/ ! 2m 2.A/ ! ! mC1.X/ ! mC1.X=A/ ! m.A/ ! 0:

A similar exact sequence exists for an arbitrary pointed map f W AX where a typical

We now generalize the suspension theorem. Let .X; / be a pointed space. Recall the suspension †X and the homomorphism † W n.X/ ! nC1.†X/.

(6.10.4) Theorem. Let X be a well-pointed space. Suppose i .X/ D 0 for 0 i n. Then † W j .X/ ! j C1.†X/ is bijective for 0 j 2n and surjective for j D 2n C 1.

Proof. Let CX D X I=.X 1[f g I / be the cone on X. We have an embedding i W X ! CX, x 7!Œx; 0 which we consider as an inclusion. The quotient CX=X can be identified with †X. From the assumption that f g X is a cofibration one concludes that i is a cofibration (Problem 1). Since CX is contractible, the exact

with the quotient map p W CX ! CX=X D †X. We can therefore prove the theorem by showing that p is bijective or surjective in the same range. This follows from the quotient theorem (6.10.2).

(6.10.5) Theorem. Let X and Y be well-pointed spaces. Assume i .X/ D 0 for i < p . 2/ and i .Y / D 0 for i < q . 2/. Then the inclusion X _ Y ! X Y induces an isomorphism of the i -groups for i p C q 2. The groups

i .X Y; X _ Y / and i .X ^ Y / are zero for i p C q 1.

Proof. We first observe that j W i .X _ Y / ! i .X Y /, induced by the inclu-

sion, is always surjective. The projections onto the factors induce isomorphisms k W i .X Y / Š i .X/ i .Y /. Let j X W X ! X _ Y and j Y W Y ! X _ Y denote the inclusions. Let

s W i .X/ i .Y / ! i .X _ Y /; .x; y/ 7!j X .x/ C j Y .y/:

Then sk is right inverse to j . Hence the exact sequence of the pair .X Y; X _ Y / yields an exact sequence

. / 0 ! iC1.X Y; X _ Y / ! i .X _ Y / ! i .X Y / ! 0:

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(6.10.7) Proposition. Let Y be k-connected .k 0/ and Z be l-connected

.l 1/ and well-pointed. Then the natural maps

j .Z/ ! j .Y Z; Y / ! j .Y Z=Y / ! j .Z/

are isomorphisms for 0 < j k C l C 1.

Proof. The first map is always bijective for j 1; this is a consequence of the exact sequence of the pair .Y Z; Y / and the isomorphism j .Y / j .Z/ Šj .Y Z/. Since the composition of the maps is the identity, we see that the second map is always injective and the third one surjective. Thus if p W j .Y Z; Y / !j .Y Z=Y is surjective, then all maps are bijective. From our assumption about Z we conclude that j .Y Z; Y / D 0 for 0 < j l (thus there is no condition for l D 0; 1). The quotient theorem now tells us that p is surjective

(6.10.9) Proposition. Suppose i .X/ D 0 for i < p . 0/ and i .Y / D 0 for i < q . 0/. Then i .X ? Y / D 0 for i < p C q C 1.

Proof. In the case that p D 0 there is no condition on X. From the definition of the join we see that X ? Y is always path connected. For p D 0 we claim thati .X ? Y / D 0 for i < q C 1. Consider the diagram

and apply (6.7.9). In the general case the excision theorem says that the map

i .CX Y; X Y / ! i .X ? Y; X C Y / is an epimorphism for i < p C q C 1. Now use diagram chasing in the diagram