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

138 Chapter 6. Homotopy Groups

(6) Let vi W W n ! R, 1 i n be functions such that vi .x/ < 0 for x 2 Ci . / and vi .x/ > 0 for x 2 Ci .C/. Then there exists x 2 W n such that vi .x/ D 0 for each i (Intermediate Value Theorem).

(7) Suppose Bi separates Ci . / and Ci .C/ for 1 i n. Then the intersection B1 \ B2 \ \ Bn is non-empty.

(8) Let B0; : : : ; Bn be a closed covering of n such that ei … Bi and @i n Bi .

Then TnD Bi D6 ;. The same conclusion holds if we assume that the Bi are

i 0

open.

and . n; @ n/, since they are homeomorphic to .Dn; Sn 1/. Statement (3) is also equivalent to the inclusion Sn 1 Rn X f0g not being null homotopic (similarly for @W n in place of Sn 1).

Proof. .1/ ) .2/. Suppose r is a retraction. Then x 7! r.x/ is a map without fixed point.

.2/ ) .3/. The map r W Dn ! Sn 1 which corresponds by (2.3.4) to a null homotopy of the identity is a retraction.

.3/ ) .1/. Suppose b has no fixed point. Then

has no fixed point, the formula for f defines a map on the whole of D , and then .x; t/ 7!f .tx/ is a homotopy from the constant map to f . Thus f is null homotopic, and therefore also id.Sn 1/.

.2/ ) .4/. If x is contained in the interior of Dn, then there exists a retraction r W Rn X x ! Sn 1 of Sn 1 Rn X x. If x is not contained in the image of f ,

to W , it is null homotopic, but the inclusion is not null homotopic. A contradiction.

of the open covering by the Ui and see that the Ui have non-empty intersection. Contradiction.

Theorem (6.6.1) has many different proofs. For a proof which uses only basic results in differential topology see [79]. Another interesting proof is based on a combinatorial result, called Sperner’s Lemma [173].

140 Chapter 6. Homotopy Groups

The retraction theorem does not hold for infinite-dimensional spaces. In [70, Chapter 19] you can find a proof that the unit disk of an infinite-dimensional Banach space admits a retraction onto its unit sphere.

Does there exist a sensible topological notion of dimension for suitable classes of spaces? Greatest generality is not necessary at this point. As an example we introduce the covering dimension of compact metric spaces X. (For dimension

theory in general see [94].) Let C be a finite covering of X and " > 0 a real number. We call C an "-covering, if each member of C has diameter less than ". We say C has order m, if at least one point is contained in m members but no point in mC1. The compact metric space X has covering dimension dim X D k, if there

exists for each " > 0 a finite closed "-covering of X of order k C 1 and k 2 N0 is minimal with this property. Thus X is zero-dimensional in this sense, if there exists for each " > 0 a finite partition of X into closed sets of diameter at most ". We verify that this notion of dimension is a topological property.

(6.6.2) Proposition. Let X and Y be homeomorphic compact metric spaces. If X is k-dimensional then also is Y .

Proof. Let h W X ! Y be a homeomorphism. Fix " > 0 and let U be the covering of Y by the open "-balls U".y/ D fx j d.x; y/ < "g. (We use d for the metrics.) Let

ıbe a Lebesgue number of the covering .h 1.U / j U 2 U/. Since dim X D k, there exists a finite closed ı-covering C of X of order k C 1. The finite closed

covering D D .h.C / j C 2 C/ of Y has then the order k C 1, and since each member of C is contained in a set h 1.U /, the covering D is an "-covering. Thus we have shown dim Y k.

We now show that dim Y k, i.e., there exists ı > 0 such that each finite closed ı-covering has order at least k C 1. Let " > 0 be a corresponding number for X. A homeomorphism g W Y ! X is uniformly continuous: There exists a ı > 0 such

that d.y1; y2/ < ı implies d.g.y1/; g.y2// < ". So if C is a ı-covering of Y , then D D .g.C / j C 2 C/ is an "-covering of X. Since D has order at least k C 1, so

(6.6.3) Proposition. There exists " > 0 such that each finite closed "-covering

.Bj j j 2 J / of n has order at least n C 1.

Proof. Let " be a Lebesgue number of the covering Ui D n X @i n, i D 0; : : : ; n.

Hence for each j 2 J there exist i such that Bj Ui , and the latter is equivalent to Bj \ @i n D ;. Suppose ek 2 Bj . Since ek 2 @i n for i 6Dk, we cannot have Bj Ui ; thus ek 2 Bj implies Bj Uk . Since each ek is contained in at

least one of the sets Bj we conclude jJ j n C 1. For each j 2 J we now choose g.j / 2 f0; : : : ; ng such that Bj \ @g.j / n D ; and set Ak D [fBj j g.j / D kg; this is a closed set because J is finite. Each Bj is contained in some Ak , hence

the Ak cover n. Moreover, by construction, Ak \ @k n D ;. We can therefore apply part (9) of (6.6.1) and find an x in the intersection of the Ak . Hence for each

142 Chapter 6. Homotopy Groups

Proof. The case n D 0 is trivial. The equivalence of (2) and (3) is a consequence of the homeomorphism .DnC1; Sn/ Š .I nC1; @I nC1/. Suppose f W Sn ! X is given. Use e1 D .1; 0; : : : / 2 Sn as a base point and think of f representing an element in n.X; x/. If (1) holds, then f is pointed null homotopic. A null homotopy Sn I ! X factors over the quotient map Sn I ! DnC1, .x; t/ 7!

.1 t/e1 C tx and yields an extension of f . Conversely, let an element ˛ ofn.X; x/ be represented by a pointed map f W .Sn; e1/ ! .X; x/. If this map has an extension F to DnC1, then .F; f / represents ˇ 2 nC1.X; X; x/ D 0 with

@ˇ D ˛.

(6.7.2) Proposition. Let n 0. Let f W .Dn; Sn 1/ ! .X; A/ be homotopic as

.x; t/ D .2˛.x; t/ 1 x; 2 ˛.x; t// with the function ˛.x; t/ D max.2kxk; 2 t/. Then H D G ı is a homotopy with the desired property from f to g D H1.

(6.7.3) Proposition. Let n 1. The following assertions about .X; A/ are equivalent:

(1) n.X; A; / D 0 for each choice of 2 A.

(2)Each map f W .I n; @I n/ ! .X; A/ is as a map of pairs homotopic to a constant map.

(3)Each map f W .I n; @I n/ ! .X; A/ is homotopic rel @I n to a map into A.

Proof. (1) ) (2). Let f W .I n; @I n/ ! .X; A/ be given. Since J n 1 is contractible, there exists a homotopy of the restriction f W J n 1 ! A to a constant map. Since

J n 1 @I n and @I n I n are cofibrations, f is as a map of pairs homotopic to g W .I n; @I n/ ! .X; A/ such that g.J n 1/ D fa0g. Since n.X; A; a0/ D 0, the

map g W .I n; @I n; J n 1/ ! .X; A; a0/ is null homotopic as a map of triples.

We call .X; A/ n-compressible if one of the assertions in (6.7.3) holds. More generally, we call a map f W X ! Y n-compressible if the following holds: For each commutative diagram

'

@I n X

\f

I n ˆ Y

there exists ‰ W I n ! X such that ‰j@I n D ' and f ‰ ' ˆ relative to @I n. (This amounts to part (3) in (6.7.3).) This notion is homotopy invariant in the following sense:

(6.7.4) Proposition. Given f W X ! Y and a homotopy equivalence p W Y ! Z. Then f is n-compressible if and only pf is n-compressible.

(6.7.5) Proposition. Let n 0. The following assertions about .X; A/ are equivalent:

(1)Each map f W .I q ; @I q / ! .X; A/, q 2 f0; : : : ; ng is relative to @I q homotopic to a map into A.

(2) The inclusion j W A ! X induces for each base point a 2 A a bijection j W q .A; a/ ! q .X; a/ for q < n and a surjection for q D n.

(3)0.A/ ! 0.X/ is surjective, and q .X; A; a/ D 0 for q 2 f1; : : : ; ng and each a 2 A.

Proof. (1) , (3). The surjectivity of 0.A/ ! 0.X/ is equivalent to (1) for q D 0. The other cases follow from (6.7.3).

(2) , (3). This follows from the exact sequence (6.1.2).

A pair .X; A/ is called n-connected if (1)–(3) in (6.7.5) hold. We call .X; A/ 1-connected if the pair is n-connected for each n. A pair is 1-connected if and

only if j W n.A; a/ ! n.X; a/ is always bijective. If X 6D ;but A D ; we say that .X; A/ is . 1/-connected, and .;; ;/ is 1-connected.

(6.7.6) Proposition. Let n 0. The following assertions about X are equivalent:

(1)q .X; x/ D 0 for 0 q n and x 2 X.

(2)The pair .CX; X/ is .n C 1/-connected.

(3)Each map f W @I q ! X, 0 q n C 1 has an extension to I q .

Proof. The cone CX is contractible. Therefore @ W qC1.CX; X; / Š q .X; /. This and (6.7.5) shows the equivalence of (1) and (2). The equivalence of (1) and

(3) uses (6.7.1).

A space X is n-connected if (1)–(3) in (6.7.6) hold for X. Note that this is compatible with our previous definitions for n D 0; 1.

144 Chapter 6. Homotopy Groups

Let f W X ! Y be a map and X Z.f / the inclusion into the mapping

cylinder. Then f is said to be n-connected if .Z.f /; X/ is n-connected. We then also say that f is an n-equivalence. Thus f is n-connected if and only if

f W q .X; x/ ! q .Y; f .x// is for each x 2 X bijective (surjective) for q < n

n.Y; f .x// is bijective for each n 0 and each x 2 X.

(6.7.7) Proposition. Let .p1; p0/ W .E1; E0/ ! B be a relative Serre fibration. Let Fjb denote the fibre of pj over b. Then the following are equivalent:

(1).E1; E0/ is n-connected.

(2).F1b; F0b/ is n-connected for each b 2 B.

The compression properties of an n-connected map can be generalized to pairs of spaces which are regular unions of cubes of dimension at most n. We use this generalization in the proof of theorem (6.7.9). Consider a subdivision of a cube I n. Let us call B a cube-complex if B is the union of cubes of this subdivision. A subcomplex A of B is then the union of a subset of the cubes in B. We understand that B and A contain with each cube all of its faces. The k-skeleton B.k/ of B consists of the cubes in B of dimension k; thus A.k/ D B.k/ \ B.

(6.7.8) Proposition. Let f W X ! Y be n-connected. Suppose .C; A/ is a pair of cube-complexes of dimension at most n. Then to each commutative diagram

there exists ‰ W C ! X such that ‰jA D ' and f ‰ ' ˆ relative to A.

Proof. Induction over the number of cubes. Let A B C such that C is obtained from B by adding a cube W of highest dimension. Then @W B. By induction there exists ‰0 W B ! X such that ‰0jA D ' and a homotopy H W f ‰0 ' ˆjB relative to A. Extend H to a homotopy of ˆ. The end ˆ1 of this homotopy satisfies ˆ1jB D f ‰0. We now use that f is n-connected and extend ‰0 over W to ‰ W C ! X such that f ‰ ' ˆ1 relative to B. Altogether we have

(6.7.9) Theorem. Let ' W .X; X0; X1/ ! .Y; Y0; Y1/ be a map such that the re-

strictions 'i W Xi ! Yi are n-connected and '01 W X0 \ X1 ! Y0 \ Y1 is .n 1/- connected. Suppose X D X0ı [ X1ı and Y D Y0ı [ Y1ı. Then ' is an n-equivalence.

Proof. We use mapping cylinders to reduce to the case of inclusions ' W X Y; 'i W Xi Yi . Let .F; f / W .I n; @I n/ ! .Y; X/ be given. We have to show that this map is homotopic relative to @I n to a map into X. Let

Ai D F 1.Y X Yiı/ [ f 1.X X Xiı/:

These sets are closed and disjoint. By the Lebesgue lemma we choose a cubical subdivision of I n such that no cube W of the subdivision intersects both A0 and A1. Let Kj be the union of the cubes W which satisfy

F .W / Yiı; f .W \ @I n/ Xiı:

We denote by K the .n 1/-skeleton of a cubical complex; then K \ @I n D

Since .Y01; X01/ is .n 1/-connected there exists a homotopy relative to @I n \ K01 from F01 to a map g01 W K01 ! X01. Define g0 W K0 \ .@I n [ K / ! X0 by

g0jK0 \ @I n D f0; g0jK0 \ K1 D g01:

(Both maps agree on the intersection.) The homotopy F01 ' g01 and the constant

constant on K0 \ @I . Since the inclusion of a cube complex into another one is a

corresponding groups (orthogonal, unitary, symplectic)

Let d D dimR F . The starting point are the (Serre) fibrations which arise from the action of the orthogonal groups on the unit spheres by matrix multiplication

i .S / D 0, i < n and the exact homotopy sequences of the fibrations we deduce that the inclusions j W O.n; F / ! O.n C 1; F / and j W SO.n; F / ! SO.n C 1; F / are d.n C 1/ 2 connected. By induction and passage to the colimit we obtain

(6.8.1) Proposition. For n < m 1, the inclusions O.n; F / ! O.m; F / and SO.n; F / ! SO.m; F / are d.n C 1/ 2 connected; in particular, the homomorphisms i .O.n; F // ! i .O.m; F // are isomorphisms in the range i n 2 .R/,

We have the corresponding (Serre) fibrations of the type H ! G ! G=H for these homogeneous spaces. We use (6.8.1) in the exact homotopy sequences of these fibrations and obtain:

The fibre over ekC1 is homeomorphic to Vk .F n/; with W v 7!.v; 0/ we obtain a homeomorphism j W .v1; : : : ; vk / 7!. v1; : : : ; vk ; ekC1/ onto this fibre. From the homotopy sequence of this fibration we obtain

(6.8.3) Proposition. j W i .Vk .F n// ! i .VkC1.F nC1// is an isomorphism for i d.n C 1/ 3.