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inclusion RjDn! Rnj0 induces a homotopy equivalence C.RnjDn/! C n; the morphisms ˛ comes from (7.2.7); and CD#f is obtained by applying the mapping cone to D#f ; ﬁnally, we multiply the homotopy class of the composition by.1/nm. We take the freedom to use.Df /nCm W C.RmjY / ^ C n! C.RnjX/ ^ C m as our model for Df, i.e., we do not compose with the h-equivalences of the type C n! S n obtained in (7.2.2). 7.4. The Complement Duality Functor 171 (7.4.2) Lemma. Let f be an inclusion, f W X Y Rn. Then Df has as a representative the map C.RnjY /! C.RnjX/ induced by the inclusion. In particular the identity of X is send to the identity. Proof. We take the scaling function y 7! kyk C1 and extend f by the identity. Then D#.f / is the map.x; y/ 7!..kyk C1/ x; y.kyk C1/ x/. The map D1 is.x; y/ 7!..kyk C1/ x; y/ and the homotopy.x; y; t/ 7!..1 t/.kyk C1/ x C t.x C y/; y/ is a homotopy of pairs from D1 to.x; y/ 7!.xCy; y/. Hence D2ıD1 is homotopic to.x; y/ 7!.x C y; x/ and the homotopy.x; y; t/ 7!..1 t/x C y; x/ shows it to be homotopic to.x; y/ 7!.y; x/. Now interchange the factors and observe that x 7! x has the degree.1/n D.1/nn. Next we consider the case of a homeomorphism f W X! Y with inverse g. Let Qg W Rm! Rn be a Tietze extension of g. Then: (7.4.3) Lemma. The maps.
x; y/ 7!.'.kykx; y/ and.x; y/ 7!.x C Qg.y/; y/ are as maps of pairs RnjDn RmjY! RnCmjG.f / homotopic. Here W Œ0; 1Œ! 0; 1Œ is a function such that.kf xk/ kxk and '.r/ D 1 C.r/. Proof. We use the linear homotopy..1 t/.x C Qg.y// C t'.kyk/ x; y/. Suppose this element is contained in G.f /. Then y 2 Y and hence g.y/ D Qg.y/, and the ﬁrst component equals g.y/. We solve for x and obtain x D t 1 C t.kyk/ g.y/: Then we take the norm kxk D t 1 C t.kyk/ kgyk t.kyk/ 1 C t.kyk/ < 1: Hence.x; y/ 2 Dn Y. In the situation of the previous lemma the map D#.f / is homotopic to the restriction of the homeomorphism Rnj0 RmjY! RnjX Rmj0 obtainable from (7.3.1). Another special case is obtained from a homeomorphism h of Rm and X Rm; Y D h.X/ Rm. In this case h and h1 are Tietze extensions. For the veriﬁcation of the functor property we start with the following data:.Rn; X/,.Rm; Y /,.Rp; Z/ and proper maps. We have the inclusion G.g/ Rm Rp and the proper map h W X! G.g/, x 7!.f x; gf x/. 172 Chapter 7. Stable Homotopy. Duality (7.4.4) Proposition. The diagram RnjDn RmjDm RpjZ nm1 RmjDm RnjDn RpjZ 1D#g 1D#.gf / RnjDn RmjY Rpj0 RmjDm RnjX Rpj0 D#f 1 RnjX Rmj
0 Rpj0 nm1 \ Rmj0 RnjX Rpj0 is homotopy commutative. Here nm are the appropriate interchange maps. Proof. For the proof we use the intermediate morphism D#.h/. In the sequel we skip the notation for the scaling function and the Tietze extension. If 'f is a scaling function for f and 'gf a scaling function for gf, then 1 W Y Z! 0; 1Œ ;.y; z/ 7! 'f.y/; 2 W Y Z! 0; 1Œ ;.y; z/ 7! 'gf.z/ are scaling functions for h. We have a factorization D2.h/ D D2 D1 we use Qh D. Qf ; gf / with gf D Qg Qf. The diagram 2.h/.x; y; z/ D.x; y Qf.x/; z/ and D2 2.h/ where 2.h/.x; y; z/ D.x; y; z gf.x// and 2.h/D1 RnCmCpjfx; 0; gf xg nm1 RmCnCpjf0; x; gf xg D2 2.h/ RnCmCpjfx; 0; 0g nm1 1D2.gf / RmCnCpjf0; x; 0g commutes. The notation f.x; 0; gf x/g means that we take the set of all element of the given form where x 2 X. We verify that the diagram RnjDn RmjDm RpjZ nm1 RmjDm RnjDn RpjZ RnCmCpjfx; 0; gf xg nm1 D1.gf / RmCnCpjf0; x; gf xg with D D1 topy. The map is, with the choice 'h D 'gf, the assignment 2h ı D1h ı.1 D1g/ and W RmjDm Rmj0 commutes up to homo-.x; y; z/ 7!.'gf.z/ x; 'g.z/ y Qf.'
gf.z/ x/; z/: We use the linear homotopy.'gf.z/ x; s.'g.z/ y Qf.'gf.z/ x/ C.1 s/y; z/. We verify that this is a homotopy of pairs, i.e., an element f Qx; 0; gf. Qx/g only occurs as the image of an element.x; y; z/ 2 Dn Dm Z. Thus assume (i) Qx D 'gf.z/ x 2 X; 7.4. The Complement Duality Functor 173 (ii) s'g.z/ y s Qf.'gf.z/ x/ C.1 s/y D 0; (iii) z D gf. Qx/. Since 'gf.z/ x 2 X, we can replace in (ii) obtain Qf by f. We apply 'gf to (iii) and 'gf.z/ D 'gf.gf.'gf.z/ x// 'gf.z/ kxk; hence kxk 1. The equation (ii) for s D 0 says y D 0, hence y 2 Dm. Thus assume s 6D 0. Then f.'gf.z/ x/ D.'g.z/ C s1 1/ y. We apply g to this equation and use (ii): z D gf.'gf.z/ x/ D g..'g.z/ C s1 1/ y/: Finally we apply 'g to this equation and obtain 'g.z/ D 'g g..'g.z/ C s1 1/ y/.'g.z/ C s1 1/kyk'g.z/kyk; and therefore kyk 1. Finally we show that the diagram RnjDn RmCpjG.g/ RnCmCpjfx; 0; 0g D#.h/ D#.f /1 1D2g RnjDn RmjY Rpj0 commutes up to homotopy. In this case we use for h the scaling function 1. Then D1h W.x; y; z/ 7!.'f.y/ x; y; z/ and.
D#f 1/ ı.1 ı D2g/.x; y; z/ D.'f.y/ x; y Qf.'f.y/ x/; z Qg.y//; D#.h/.x; y; z/ D.'f.y/ x; y Qf.'f.y/ x/; z Qg Qf.'f.y/ x//: Again we use a linear homotopy with z Qg..1 t/y C t Qf.'f.y/ x// as the third component and have to verify that it is a homotopy of pairs. Suppose the image is contained in fx; 0; 0g. Then (i) 'f.y/ x 2 X; (ii) y D Qf.'f.y/ x/ (iii) z D Qg..1 t/y C tf.'f.y/ x//.i/ D f.'f.y/ x/ 2 Y ;.ii/ D g.y/. (iii) shows that.y; z/ 2 G.g/. We apply 'f to (ii) and see that kxk 1. The three diagrams in this proof combine to the h-commutativity of the diagram in (7.4.4). (7.4.5) Proposition. Suppose that h W X I! Y is a proper homotopy. Then D.h0/ D D.h1/. 174 Chapter 7. Stable Homotopy. Duality Proof. Let j0 W X! X I, x 7!.x; 0/. The map is the composition of the homeomorphism a W X! X 0 and the inclusion b W X 0 X I. Thus Da is an isomorphism and Db is induced by the inclusion RnC1jX I RnC1jX 0; hence Db is an isomorphism, since induced by a homotopy equivalence (use (7.4.2)). Thus Dj0 is an isomorphism. The composition pr ıj0 is the identity; hence D.pr/ is inverse to D.j0/. A similar argument for j1 shows that Dj0 D Dj1. We conclude that the maps ht D h ı jt have the same image under D. (7.4.6) Rem
ark. The construction of the dual morphism is a little simpler for a map between compact subsets of Euclidean spaces. Let X Rn be compact. Choose a disk D such that X D. Then the dual morphism is obtained from Rnj0 RmjY RnjD RmjY RnCmjG.f /! RnjX Rmj0 where the last morphism is as before.x; y/ 7!.x; y Qf.x//. Also the proof of the functoriality (7.4.4) is simpler in this case. The composition.D#f 1/.1 D#g/ is.x; y; z/ 7!.x; y Qf.x/; z Qg.y//. The other composition is.x; y; z/ 7!.x; y; z Qg Qf.x//. Then we use the homotopies of pairs.x; y Qf.x/; z Qg..1t/yCt Qf.x/// and.x; yt Qf.x/; z Qg Qf.x//. Þ Problems 1. Verify in detail that the commutativity of the diagram in (7.4.4) implies that D is a functor. 2. Use the homotopy invariance of the duality functor and generalize (7.1.3) as follows. Suppose X Rn and Y Rm are closed subsets which are properly homotopy equivalent. Let n m. (1) If Rn X X 6D ;, then Rm X Y 6D ;: (2) Let Rn 6D X. For each choice of a base point Rm X Y has the same stable homotopy type as †nm.Rn X X/. (3) If Rn X X is empty and Rm X Y is non-empty, then Rm X Y has the stable homotopy type of S mn1. (4) If n D m then the complements of X and Y have the same number of path components. 3. Let X Rn and Y Rm be closed subsets and f W X! Y a proper map. Consider the closed subspace W Rm R Rn of points (.y; t; x/ D y 2 Y; t D 0
; x D 0..1 t/f.x/; t; tx/; x 2 X; t 2 I: Then W is homeomorphic to the mapping cylinder Z.f / of f. 4. Let X Rn and f W X! Rn Rm the standard embedding x 7!.x; 0/. Then Df is represented by the homotopy equivalence C.RnjX Rmj0/ ˛ C.RnjX/ ^ C m'C.RnjX/ ^ S m: 7.5. Duality 175 (Direct proof or an application of (7.4.3).) 5. Let X Rn be compact. Suppose kxk r > 0 for x 2 X. Then the constant function '.t/ D r is a scaling function for each f W X! Y. Show that the map C.RmjY / ^ C n! C n ^ C.RmjY /! C.RnjX/ ^ C m which is obtained from the deﬁnition in (7.4.6) is homotopic to the map C.D#f / of the general deﬁnition. 7.5 Duality We have associated to a proper map between closed subsets of Euclidean spaces a dual morphism in the stable category ST. If X Rn then the stable homotopy type of Rn X X or C.RnjX/ is to be considered as a dual object of X. There is a categorical notion of duality in tensor categories. Let A and B be pointed spaces. An n-duality between.A; B/ consists of an evaluation and a coevaluation such that the following holds: (1) The composition " 1 ^ "/. ^ 1 is homotopic to the interchange map. (2) The composition." ^ 1/. is homotopic to.1/n. We now construct an n-duality for.B; A/ D.C.RnjK/; KC/ where K Rn is a suitable space. In the general deﬁnition of an n-duality above we now replace S n by C n. The evaluation is deﬁned to be " W C.RnjK/ ^ C.K; ;/! C.RnjK KjK/ C
.d /! C.Rnj0/ where d is the difference map d W.Rn K;.Rn X K/ K/!.Rn; Rn X 0/;.x; k/ 7! x k as a map of pairs. This deﬁnition works for arbitrary K Rn. 176 Chapter 7. Stable Homotopy. Duality Let K Rn be compact and D Rn a large disk which contains K. Let V be an open neighbourhood of K. Consider the following diagram Rnj0 V i RnjD RnjK V jV RnjK r1 V jV RnjK j [ V jK with the diagonal W x 7!.x; x/. We apply the mapping cone functor and (7.2.1), (7.2.7). The maps i and j induce h-equivalences. We obtain V W C n! V C ^ C.RnjK/: We want to replace V by K in order to obtain the desired map. This can be done if we assume that there exists a retraction r W V! K of K V. Then we can compose with r C W V C! KC and obtain a coevaluation W C n! C.K; ;/ ^ C.RnjK/: We call a closed subspace K Rn a Euclidean neighbourhood retract (D ENR) if there exists a retraction r W V! K from a suitable neighbourhood V of K in Rn. We mention here that this is a property of K that does not depend on the particular embedding into a Euclidean space; see (18.4.1). The basic duality properties of " and are: (7.5.1) Proposition. The maps " and are an n-duality for the pair.KC; C.RnjK//. Proof. For the proof of the ﬁrst assertion we consider the diagram RnjD KjK RnjK KjK j 1 V jK KjK ˛ ˇ 1 V jV Rnj0 1d V jV RnjK KjK with ˛.x; y/ D.y; x/, ˇ.x; y/ D.y; x y/, and D.1 d /. 1/ W
.x; y/ 7!.x; x y/. The homotopy.x; y; t/ 7!.tx C.1 t/y; x y/ shows that the right square is h-commutative and the homotopy.x; y; t/ 7!.y; x ty/ shows that the triangle is h-commutative. The axiom (1) of an n-duality now follows if we write out the morphisms according to their deﬁnition and use the result just proved. For the proof of the axiom (2) we start with the diagram RnjK RnjD RnjK RnjK 1j RnjK V jK 1j 0 ˛ ˇ.1r1/.1/ RnjD RnjK d 1 RnjK KjK RnjK 7.5. Duality 177 with ˛.x; y/ D.y; x/, ˇ.x; y/ D.x y; x/, and.x; y/ D.x y; y/. The homotopy.x; y; t/ 7!.x ty.1 t/r.y/; y/ shows that the bottom triangle is hcommutative; the homotopy.x; y; t/ 7!.x y;.1t/x Cty/ shows'ˇ.1j /; the homotopy.x; y; t/ 7!.tx y; x/ shows ˛'ˇ.1 j 0/. Again we write out the morphisms according to their deﬁnition and use this result. Given a natural duality for objects via evaluations and coevaluations one can deﬁne the dual of an induced map. We verify that we recover in the case of compact ENR the morphisms constructed in the previous section. The following three proposition verify that the duality maps have the properties predicted by the categorical duality theory. (7.5.2) Proposition. Let X Rn be compact and a retract of a neighbourhood V. The following diagram is homotopy commutative C.RmjY / ^ C n 1^X C.RmjY / ^ C.X; ;/ ^ C.RnjX/.Dfm
Cn/ 1^C.f /^1 C m ^ C.RnjX/ "Y ^1 C.RmjY / ^ C.Y; ;/ ^ C.RnjX/: Proof. We reduce the problem to maps of pairs. We use the simpliﬁed deﬁnition (7.4.6) of the duality map. First we have the basic reduction RmjY Rnj0 RmjY RnjD! RmjY RnjX RmjY V jX: Then the remaining composition RmjY V jX! Rmj0 RnjX which involves, C.f /, " is the assignment.y; x/ 7!.y f r.x/; x/. The other map is.y; x/ 7! Qf jV D f r (by possibly.y Qf.x/; x/. Now we observe that we can arrange that passing to a smaller neighbourhood, see Problem 1). Dual maps are adjoint with respect to evaluation and coevaluation. This is the content of (7.5.3) and (7.5.4). (7.5.3) Proposition. The following diagram is homotopy commutative C.RnjD/ ^ C.RmjY / ^ C.X; ;/ D#f ^1 C.RnjX/ ^ C.Rmj0/ ^ C.X; ;/ 1^1^C.f / C.RnjD/ ^ C.RmjY / ^ C.Y; ;/ i^"."^1/ C n ^ C m: Proof. Consider the diagram RnjDn RmjY XjX D#f 1 RnjX Rmj0 XjX 11f.d 1/ RnjDn RmjY Y jY id Rnj0 Rmj0: 178 Chapter 7. Stable Homotopy. Duality The composition down-right sends the element.a; b; x/ to.a; b f.x// and the composition right-down to.'.b/ a x; b Qf.'.b/ a//. We use the homotopies.'.b/ax; b Qf.t'.b/a
C.1t/x// and then..1t/aCt.'.b/ax/; bf.x//. (7.5.4) Proposition. Let X and Y be compact and retracts of open neighbourhoods. Then the following diagram is homotopy commutative C n ^ C m.1^/ ^1 C.X; ;/ ^ C.RnjX/ ^ C m C.f /^1^1 C.Y; ;/ ^ C.RnjD/ ^ C.RmjY / 1^D#f C.Y; ;/ ^ C.RnjX/ ^ C m: Proof. We unravel the deﬁnitions and deform suitable maps between pairs. The composition.1 ^ D#f /. ^ 1/.1 ^ / is induced by maps Rnj0 Rmj0 RnjD RmjD RnjD RmjY Y jY RnjX Rmj0 ˛ RnjD W jY with ˛.x; y/ D.rY.y/; x; y Qf.x//. Further investigations concern ˛. We use the next diagram RnjD W jY RnjX W jY V jX W jY! ˛ ˛ ˛ Y jY RnjX Rnj0 ˛.V W /jG.f / U jG.f /: Let W V W I! Rm;.x; y; t/ 7! ty C.1 t/f rX.x/: This homotopy is constant on G.f /. Hence there exists an open neighbourhood U of G.f / such that.U I / W. On U we consider the homotopy of ˛ given by.rY.ty C.1 t/f rX.x//; x; y Qf.x//. For t D 0 we obtain the morphism.f rX.x/; x; y Qf.x// which is deﬁned on V jX RmjY. Consider the composition.C.f / ^ 1 ^ 1/. ^ 1/. It is induced by V jX Rmj0! Y jY RnjX Rmj0;.x; y
/ 7!.f rX.x/; x; y/: Now we use the homotopy.f rX.x/; x; y t Qf.x//. For t D 1 this homotopy is deﬁned on V jX RmjY.! 7.6. Homology and Cohomology for Pointed Spaces 179 (7.5.5) Remark. Let X Rn be a compact ENR and f W X! X a continuous map. From the associated n-duality we obtain a homotopy class f W S n! X C ^ C.RnjX/! C.RnjX/ ^ X C 1^f C! C.RnjX/ ^ X C "! S n: The degree d.f / D L.f / 2 Z is an interesting invariant of the map f, the Lefschetz ﬁxed point index. If f is the identity, then L.id/ is the Euler characteristic Þ of X. [51] [54] Problems 1. Let A be a closed subset of a normal space X. Let r W W! A be a retraction of an open neighbourhood. Choose open sets U; V such that A U xU V xV W: Choose a continuous function'W X! Œ0; 1 such that '.U / D f1g and '.X X V / D f0g. Let f W A! Œ0; 1 be continuous. Deﬁne F W X! Œ0; 1 by F.x/ D '.x/ f r.x/ for x 2 xV and F.x/ D 0 for x 2 X X V. Then F is a Tietze extension of f and F jU D f rjU. 2. Verify directly that the homotopy class of the coevaluation does not depend on the choice of the retraction r W V! X. 3. The n-dualities which we have constructed can be interpreted as representative elements for morphisms in the category ST. We obtain " W.C.RnjX/; n/ ˝.X C; 0/!.S 0; 0/ W.S 0; 0/!.X C; 0/ ˝.C.RnjX/; n/: They satisfy
the relations.1 ^ "/. ^ 1/ D id;." ^ 1/.1 ^ / D id which deﬁne dualities in tensor categories. 7.6 Homology and Cohomology for Pointed Spaces A homology theory for pointed spaces with values in the category R-MOD of left modules over the commutative ring R consists of a family. Qhn j n 2 Z/ of functors Qhn W TOP0! R- MOD and a family..n/ j n 2 Z/ of natural suspension isomorphisms D.n/ W Qhn! QhnC1 ı †. These data are required to satisfy the following axioms. (1) Homotopy invariance. For each pointed homotopy ft the equality Qhn.f0/ D Qhn.f1/ holds. (2) Exactness. For each pointed map f W X! Y the induced sequence Qhn.C.f // is exact. Qhn.Y / Qhn.X/ f1 f 180 Chapter 7. Stable Homotopy. Duality Let.Xj j j 2 J / be a family of well-pointed spaces with inclusions i W X! W j 2J Xj of the summands. The theory is called additive, if L j 2J Qhn.Xj /! Qhn W j 2J Xj ;.xj / 7! P j 2J.ij /.xj / is always an isomorphism. As a variant of the axioms we require the suspension isomorphisms and the exact sequences only for well-pointed spaces. If we apply the exactness axiom to the identity of a point P we see that Qhn.P / D 0. If X and Y are well-pointed, then the inclusion and projection give a coﬁbre sequence X! X _ Y! Y. This is used to verify that the additivity isomorphism holds for a ﬁnite number of well-pointed spaces. The groups Qhn.S 0/ are the coefﬁcient groups of the theory. A natural transformation of homology theories for pointed spaces consists of a family of natural transformations Qhn./! Qkn./ which commute with the suspension isomorphisms. A cohomology theory for pointed spaces consists of a family of contravariant functors Qhn W TOP0! R- MOD
and natural suspension isomorphisms D.n/ W Qhn! QhnC1 ı † such that the analogous axioms (1) and (2) hold. The theory is called additive, if W j 2J Xj! Q j 2J Qhn Qhn.Xj /; x 7!..ij /.x// is always an isomorphism for well-pointed spaces Xj. In Chapter 10 we deﬁne homology theories by the axioms of Eilenberg and Steenrod. They involve functors on TOP.2/. We show in Section 10.4 that they induce a homology theory for pointed spaces as deﬁned above. Given a homology theory Qh for pointed spaces we construct from it a homology theory for pairs of spaces as follows. We set hn.X; A/ D Qhn.C.X; A//. It should be clear that the hn are part of a homotopy invariant functor TOP.2/! R- MOD. We deﬁne the boundary operator as the composition @ W hn.X; A/ D Qhn.C.i C// p.i/ Qhn.†.AC// Š Qhn1.AC/ D hn1.A/: The isomorphism is the given suspension isomorphism of the theory Qh. The Eilenberg–Steenrod exactness axioms holds; it is a consequence of the assumption that Qh transforms a coﬁbre sequence into an exact sequence and of the naturality of the suspension isomorphism. The excision isomorphism follows from (7.2.5). We need the additional hypothesis that the covering is numerable. Remark (7.2.6) is relevant for the passage from one set of axioms to the other. 7.7. Spectral Homology and Cohomology 181 7.7 Spectral Homology and Cohomology In this section we report about the homotopical construction of homology and cohomology theories. We work in the category of compactly generated spaces. A pre-spectrum consists of a family.Z.n/ j n 2 Z/ of pointed spaces and a family.en W †Z.n/! Z.n C 1/ j n 2 Z/ of pointed maps. Since we only work with pre-
spectra in this text, we henceforth just call them spectra. A spectrum is called an -spectrum, if the maps "n W Z.n/! Z.n C 1/ which are adjoint to en are pointed homotopy equivalences. Let Z D.Z.n/; "n/ be an -spectrum. We deﬁne Zn.X/ D ŒX; Z.n/0 for a pointed space X. Since Z.n/ is up to h-equivalence a double loop space, namely Z.n/'2Z.n C 2/, we see that Zn.X/ is an abelian group, and we can view Zn as a contravariant and homotopy invariant functor TOP0! Z- MOD. We deﬁne W Zn.X/ Š ZnC1.†X/ via the structure maps and adjointness as ŒX; Z.n/0."n/ ŒX; Z.n C 1/0 Š Œ†X; Z.n C 1/0: We thus have the data for a cohomology theory on TOP0. The axioms are satisﬁed (Puppe sequence). The theory is additive. We now associate a cohomology theory to an arbitrary spectrum Z D.Z.n/; en/. For k 0 we have morphisms n W Œ†kX; Z.n/0 † bk Œ†.†kX/; †Z.n/0.en/ Œ†kC1X; Z.n C 1/0 : Let Znk.X/ be the colimit over this system of morphisms. The bk n are compatible with pointed maps f W X! Y and induce homomorphisms of the colimit groups. In this manner we consider Zn as a homotopy invariant, contravariant functor TOP0! Z- MOD. n are for k 2 homomorphisms between abelian groups.) The exactness axiom again follows directly from the coﬁbre sequence. The suspension isomorphism is obtained via the identity (The bk Œ†kC1X; Z.n C k C 1/0 Š Œ†k.†X/; Z.n C k
C 1/0 which gives in the colimit Zn.X/ Š ZnC1.†X/. If the spectrum is an -spectrum, we get the same theory as before, since the canonical morphisms ŒX; Z.n/0! Zn.X/ are natural isomorphisms of cohomology theories. Because of the colimit process we need the spaces Z.k/ only for k k0. We use this remark in the following examples. 7.7.1 Sphere spectrum. We deﬁne Z.n/ D S.n/ and en W †S.n/ Š S.n C 1/ the identity. We set!k.X/ D colimnŒ†nX; S nCk0 and call this group the k-th stable Þ cohomotopy group of X. 182 Chapter 7. Stable Homotopy. Duality 7.7.2 Suspension spectrum. Let Y be a pointed space. We deﬁne a spectrum with Þ spaces †nY and en W †.†nY / Š †nC1Y. 7.7.3 Smash product. Let Z D.Z.n/; en/ be a spectrum and Y a pointed space. The spectrum Y ^ Z consists of the spaces Y ^ Z.n/ and the maps id ^en W †.Y ^ Z.n// Š Y ^ †Z.n/! Y ^ Z.n C 1/: (Note that †A D A^I =@I. Here and in other places we have to use the associativity of the ^-product. For this purpose it is convenient to work in the category of k-spaces.) We write in this case Zk.XI Y / D colimnŒ†nX; Y ^ Z.n C k/0: The functors Zk.I Y / depend covariantly on Y : A pointed map f W Y1! Y2 induces a natural transformation of cohomology theories Zk.I Y1/! Zk.I Y2/. Þ In general, the deﬁnition of the cohomology theory Z./ has to be improved, since this theory may not be additive. We now construct homology theories. Let E D.E.n/; en W
E.n/ ^ S 1! E.n C 1/ j n 2 Z/ be a spectrum. We use spheres as pointed spaces and take as standard model the one-point compactiﬁcation S n D Rn [ f1g. If V is a vector space one often writes S V D V [ f1g with base point 1. Then we have a canonical homeomorphism S V ^ S W Š S V ˚W, the identity away from the base point. A linear isomorphism f W V! W induces a pointed map S f W S V! S W. The homology group Ek.X/ of a pointed space X is deﬁned as colimit over the maps b W ŒS nCk; X ^E.n/0! ŒS nCk^S 1; X ^E.n/^S 10! ŒS nCkC1; X ^E.nC1/0: The ﬁrst map is ^S 1 and the second map is induced by id ^en. For n C k 2 the morphism b is a homomorphism between abelian groups. It should be clear from the deﬁnition that Ek./ is a functor on TOP0. We need the suspension morphisms. We ﬁrst deﬁne suspension morphisms l W Ek.Z/! EkC1.S 1 ^ Z/: They arise from the suspensions S 1 ^ ŒS nCk; Z ^ E.n/0! ŒS nCkC1; S 1 ^ Z ^ E.n/0; which are compatible with the maps b above. Then we set D.1/kl where the map interchanges the factors. 7.7. Spectral Homology and Cohomology 183 (7.7.4) Lemma. l is an isomorphism. Proof. Let x 2 Ek.Z/ be contained in the kernel of l. Then there exists Œf 2 ŒS nCk; Z ^ E.n/0 representing x such that 1 ^ f is null homotopic. Consider the diagram S 1 ^ S nCk 1 S nCk ^ S 1 1^f f ^1 S 1 ^ Z ^ E.n/ 2 Z ^ E.n/ ^ S 1 1^e Z ^
E.n C 1/ with permutation of factors 1 and 2. Since 1 ^ f is null homotopic, the representative.1 ^ e/ ı.f ^ 1/ of x is also null homotopic. This shows that l is injective. In order to prove surjectivity we consider the two-fold suspension. Let x 2 EkC2.S 2 ^ Z/ have the representative g W S nCkC2! S 2 ^ Z ^ E.n/. Then f W S nCkC2 g! S 2 ^ Z ^ E.n/! Z ^ E.n/ ^ S 2 e2! Z ^ E.n C 2/ represents an element y 2 Ek.Z/. Here e2 is the composition of the spectral structure maps E.n/ ^ S 2 D.E.n/ ^ S 1/ ^ S 1! E.n C 1/ ^ S 1! E.n C 2/: We show 2 surjective and injective and hence the same holds for the ﬁrst l. l.y/ D x. Once we have proved this we see that the second l is The proof of the claim is based on the next diagram with interchange maps ; 0; 00. S 2 ^ S nCkC2 1^g 00 00 1 S nCkC2 ^ S 2 g^1 S 2 ^.S 2 ^ Z ^ E.n// S 2 ^ Z ^ E.n/ ^ S 2 " 0 1^ S 2 ^.Z ^ E.n/ ^ S 2/ 1^e2 1^e2 S 2 ^ Z ^ E.n C 2/ D S 2 ^ Z ^ E.n C 2/ The maps 0 and 00 are homotopic to the identity, since we are interchanging a sphere with an even-dimensional sphere. The composition of the left verticals represents 2 l.y/, and the composition of the right verticals represents x. " 184 Chapter 7. Stable Homotopy. Duality (7.7.5) Proposition. For each pointed map f W Y! Z the sequence Ek.Y / f Ek.Z/ f1 Ek.C.f // is exact. Proof. The exactness is again a simple consequence of the coﬁbre sequence. But since the coﬁbre sequence is inserted into the “wrong” covariant part
, passage to the colimit is now essential. Suppose z 2 Ek.Z/ is contained in the kernel of f1. Then there exists a representing map h W S nCk! Z ^ E.n/ such that.f1 ^ 1/ ı h is null homotopic. The next diagram compares the coﬁbre sequences of id W S nCk! S nCk and f ^ 1 W Y ^ E.n/! Z ^ E.n/. S nCk h id1 C.id/ H Z ^ E.n/ C.f ^ 1/.f ^1/1 # f1^1 p.id/ S nCk ^ S 1 id S nCk ^ S 1 ˇ h^1 p.f ^1/ Y ^ E.n/ ^ S 1 f ^1^1 Z ^ E.n/ ^ S 1'1^e 1^e C.f / ^ E.n/ Y ^ E.n C 1/ f ^1 Z ^ E.n C 1/ The map'is the canonical homeomorphism (in the category of k-spaces) which makes the triangle commutative. Since.f ^ 1/1 ı h is null homotopic, there exists H such that the ﬁrst square commutes. The map ˇ is induced from.h; H / by passing to the quotients, therefore the second square commutes. It is a simple consequence of the earlier discussion of the coﬁbre sequence that the third square is h-commutative (Problem 1). The composition.1^e/.h^1/ is another representative of z, and the diagram shows that.1 ^ e/ˇ represents an element y 2 Ek.Y / such that fy D z. A similar proof shows that the Zk.XI Y / form a homology theory in the vari- able Y. Problems 1. Consider the coﬁbre sequences of two maps f W A! B and f 0 W A0! B 0. diagram In the A A0 f f 0 f1 f 0 1 B h B 0 C.f / p.f / †A †f †B H ˇ †h C.f 0/ p.f 0/ †A0 †f 0 †B 0 # 7.8
. Alexander Duality 185 assume given h and H such that the ﬁrst square commutes. The map ˇ is induced from.h; H / by passing to the quotients. Show that the third square commutes up to homotopy (use (4.6.2)). 2. Show that the homology theory deﬁned by a spectrum is additive (for families of wellpointed spaces). 3. Show that a weak pointed h-equivalence between well-pointed spaces induces an isomorphism in spectral homology. The use of k-spaces is therefore not essential. 4. Let Z be the sphere spectrum (7.7.1). Then, in the notation of (7.7.3), ST..X; n/;.Y; m// D Zmn.XI Y /; the morphism set of the category ST of Section 7.1. 7.8 Alexander Duality Let E D.En; e.n/ W En ^ S 1! EnC1/ be a spectrum. Let W S n! B ^ A, " W A ^ B! S n be an n-duality. The compositions ŒA ^ S t ; EkCt 0 B^ ŒB ^ A ^ S t ; B ^ EkCt 0 ŒS n ^ S t ; B ^ EkCt 0 are compatible with the passage to the colimit and induce a homomorphism D W E k.A/! Enk.B/: The compositions ŒS tCnk; B ^ Et 0 ^A ŒA ^ S tCnk; A ^ B ^ Et 0 ŒA ^ S nCkt ; Et ^ S n0 " e ŒA ^ S nCkt ; S n ^ Et 0 ŒA ^ S tCnk; EtCn0 with the interchange map are compatible with the passage to the colimit if we multiply them by.1/nt. They induce a homomorphism D W Enk.B/! Ek.A/: (7.8.1) Theorem (Alexander duality). The morphisms D and D are isomorphisms. They satisfy DD D.1/nk id and DD D.1/nk id. Proof. The relations of the theorem are a direct consequence of the deﬁning properties of an n-duality. The composition "
ı.A ^ / ı ı.B ^ / equals ı†n W ŒA^S t ; EkCt 0! ŒS n^A^S t ; S n^EkCt 0! ŒA^S n^S t ; S n^EkCt 0: The deﬁnition of D then involves the sign.1/n.kCt/. This morphism differs from a map in the direct system for Ek.A/ by the interchange, hence by a sign.1/nt. Hence the sign.1/nk remains. The second relation is veriﬁed similarly. 186 Chapter 7. Stable Homotopy. Duality Let PE./ and P h./ be the homology and cohomology theories on TOP.2/ constructed from the theories E./ and E./. If we use the n-duality between X C and C.RnjX/ for a compact ENR in Rn we obtain isomorphisms PEnk.Rn; Rn X X/ Š PEk.X/; PEnk.X/ Š PE.Rn; Rn X X/: This is the usual appearance of Alexander duality. In this setting one can also work with the bi-variant theory Zk.XI Y / D Zk.XI Y /. Then one obtains from an n-duality an adjointness isomorphism Zk.A ^ XI Y / Š Znk.XI B ^ Y /. A homology theory h./ is deﬁned on the category ST. Here one deﬁnes hl..X; n// D hln.X/. Let a morphism f 2 ST..X; n/;.Y; m// be represented by fk W X ^ S nCk! Y ^ S mCk. The induced morphism is deﬁned by commutativity of the next diagram hl..X; n// D hln.X/ †nCk hlCk.X ^ S nCk/ hl.f / hl..Y; m// D hlm.Y / †mCk hlCk.Y ^ S mCk/:.fk / Given a homology theory h./ one can deﬁne via the
complement duality functor a sort of cohomology for spaces which admit an embedding as a closed subset of a Euclidean space. Let X be such a space. Choose an embedding i W X! Rn and deﬁne hk.iX/ D hnk.C.RnjiX// D hk.C.RnjiX/; n/. If j W X! Rm is another embedding, we have the homeomorphism j i 1 W iX! jY and we have the duality map D.j i 1/. The set of embeddings together with the morphisms D.j i 1/ from i to j form a contractible groupoid; it is a complicated replacement for the space X. We obtain the induced contractible groupoid of the hk.iX/. It is equivalent to a group which we denote hk.X/. From the complement duality functor we obtain a well-deﬁned homomorphism hk.f / W hk.Y /! hk.X/ for a proper map f W X! Y ; in this way hk./ becomes a contravariant functor. We do not discuss in what sense the hk.X/ can be made into a cohomology theory. This cohomology theory is the “correct” one for duality theory in the sense that the Alexander duality isomorphism hk.X/ Š hk.C.RnjX/; n/ holds for all spaces in question (and not only for compact ENR). A similar devise can be applied to a given cohomology theory. One obtains a homology theory which is again the “correct” one for duality theory. 7.9 Compactly Generated Spaces Several constructions in homotopy theory lead to problems in general topology. A typical problem arises from the fact that a product of quotient maps is in general 7.9. Compactly Generated Spaces 187 no longer a quotient map. We met this problem already in the discussion of CWcomplexes. In this auxiliary section we report about some devices to deal with such problems. The idea is to construct a category with better formal properties. One has to pay a price and change some of the standard notions, e.g., redeﬁne topological products. A compact Hausdorff space will be called
a ch-space. For the purpose of the following investigations we also call a ch-space a test space and a continuous map f W C! X of a test space C a test map. A space X is called weakly hausdorff or wh-space, if the image of each test map is closed. (7.9.1) Proposition. A Hausdorff space is a wh-space. A wh-space is a T1-space. A space X is a wh-space if and only if each test map f W K! X is proper. If X is a wh-space, then the image of each test map is a Hausdorff space. A subspace of a wh-space is a wh-space. Products of wh-spaces are wh-spaces. A subset A of a topological space.X; T / is said to be k-closed (k-open), if for each test map f W K! X the pre-image f 1.A/ is closed (open) in K. The k-open sets in.X; T / form a topology kT on X. A closed (open) subset is also k-closed (k-open). Therefore kT is ﬁner than T and the identity D X W kX! X is continuous. We set kX D k.X/ D.X; kT /. Let f W K! X be a test map. The same set map f W K! kX is then also continuous. For if U kX is open, then U X is k-open, hence f 1.U / K is open. Therefore X induces for each ch-space K a bijection. TOP.K; kX/ Š! TOP.K; X/; f 7! X ı f: Hence X and kX have the same k-open sets, i.e., k.kX/ D kX. A topological space X is called k-space, if the k-closed sets are closed, i.e., if X D kX. Because of k.kX/ D kX the space kX is always a k-space. A k-space is also called compactly generated. We let k-TOP be the full subcategory of TOP with objects the k-spaces. A whk-space is a space which is a wh-space and
a k-space. The next proposition explains the deﬁnition of a k-space. We call a topology S on X ch-deﬁnable, if there exists a family.fj W Kj! X j j 2 J / of test maps such that: A X is S-closed, for each j 2 J the pre-image f 1.A/ is closed in ` j j Kj!.X; S/ Kj. We can rephrase this condition: The canonical map hfj i W is a quotient map. A ch-deﬁnable topology is ﬁner than T. We deﬁne a partial ordering on the set of ch-deﬁnable topologies by S1 S2, S1 S2. (7.9.2) Proposition. The topology kT is the maximal ch-deﬁnable topology with respect to the partial ordering. Proof. By Zorn’s Lemma there exists a maximal ch-deﬁnable topology S. If this topology is different from kT, then there exists an S-open set U, which is not k-open. Hence there exists a test map t W K! X such that t 1.U / is not open. If 188 Chapter 7. Stable Homotopy. Duality we adjoin this test map to the deﬁning family of S, we see that S is not maximal. (7.9.3) Corollary. The k-spaces are the spaces which are quotients of a topological sum of ch-spaces. (7.9.4) Proposition. The following are equivalent: (1) X is a k-space. (2) A set map f W X! Y is continuous if and only if for each test map t W K! X the composition f t is continuous. Proof. (1) ) (2). Let U Y be open. In order to see that f 1.U / is open it sufﬁces to show that this set is k-open, since X is a k-space. Let t W K! X be a test map and f t continuous. Then k1.f 1.U // is open, and this shows what we want. (2) ) (1). We show that the identity
X! kX is continuous. This holds by (2) and because X and kX have the same test maps. (7.9.5) Proposition. Let f W X! Y be continuous. Then the same set map kf W kX! kY is continuous. The assignments X 7! kX, f 7! kf yield a functor k; moreover, we have the inclusion functor i, k W TOP! k- TOP; i W k- TOP! TOP : (7.9.6) Proposition. The functor k is right adjoint to the functor i. Proof. A natural bijection is k- TOP.Y; kX/ Š TOP.iY; X/, f 7! ı f. This If Y is a k-space and f W Y! X continuous, then map is certainly injective. kf W Y D kY! kX is continuous; this is used to show surjectivity. (7.9.7) Proposition. Let X be a wh-space. Then A X is k-closed if and only if for each ch-space K X the set A \ K is closed in K. In particular a wh-space X is a k-space if and only if: A X is closed, for each ch-space K X the intersection A \ K is closed in K. Proof. Let A be k-closed. The inclusion K X of a ch-space is a test map. Hence A \ K is closed in K. Conversely, suppose that A satisﬁes the stated condition and let f W L! X be a test map. Since X is a wh-space, f.L/ is a ch-space and therefore f.L/ \ A is closed in f.L/. Then f 1.A/ D f 1.f.L/ \ A/ is closed in L D f 1f.L/. This shows: A is k-closed. Thus we see that wh-spaces have an internal characterization of their k-closed sets. For wh-spaces therefore k.X / can be deﬁned from internal properties of X. If X is a wh-space, so is kX. 7.9. Compactly Generated Spaces 189 (7.9.8) Theorem. X is a k-space under one of the following conditions: (1) X is metrizable
. (2) Each point of X has a countable neighbourhood basis. (3) Each point of X has a neighbourhood which is a ch-space. (4) For Q X and x 2 xQ there exists a ch-subspace K X such x is contained in the closure of Q \ K in K. (5) For each Q X the following holds: Q \ K open (closed) in K for each test space K X implies Q open (closed) in X. Proof. (1) is a special case of (2). (2) Let Q X and suppose that f 1.Q/ is closed for each test map f W C! X. We have to show that Q is closed. Thus let a 2 xQ and let.Un j n 2 N/ be a neighbourhood basis of a. For each n choose an 2 Q \ U1 \ \ Un. Then the sequence.an/ converges to a. The subspace K D f0; 1; 21; 31; : : :g of R is compact. The map f W K! X, f.0/ D a, f.n1/ D an is continuous, and n1 2 f 1.Q/. By assumption, f 1.Q/ is closed in K, hence 0 2 f 1.Q/, and therefore a D f.0/ 2 Q..3/ ).4/. Let Q X and suppose a 2 xQ. We choose a ch-neighbourhood K of a and show that a is contained in the closure of Q \ K in K. Thus let U be a neighbourhood of a in K. Then there exists a neighbourhood U 0 of a in X such that U 0 \ K U. Since U 0 \ K is a neighbourhood of a in X and a 2 xQ, we conclude U \.Q \ K/.U 0 \ K/ \.Q \ K/ D.U 0 \ K/ \ Q ¤ ;: Hence a is contained in the closure of Q \ K in K..4/ ).5/. Suppose Q \ K is closed in K for every test subspace K X. Let a 2 xQ. By (4), there exists a test subspace K0 of X, such that a is contained in the closure of Q \ K0 in K0. By the assumption (5), Q \ K0 is closed in K0; and hence a 2 Q \ K0
Q. (5) Let f 1.Q/ be closed in K for each test map f W K! X. Then, in particular, for each test subspace L X the set Q \ L is closed in L. The assumption (5) then says that Q is closed in X. This shows that X is a k-space. (7.9.9) Theorem. Let p W Y! X be a quotient map and Y a k-space. Then X is a k-space. Proof. Let B X be k-closed. We have to show that B is closed, hence, since p is a quotient map, that p1.B/ is closed in Y. Let g W D! Y be a test map. Then g1.p1.B// D.pg/1.B/ is closed in D, because B is k-closed. Since Y is a k-space, p1.B/ is closed in Y. 190 Chapter 7. Stable Homotopy. Duality (7.9.10) Proposition. A closed (open) subspace of a k-space is a k-space. The same holds for whk-spaces. Proof. Let A be closed and B A a subset such that f 1.B/ is closed in C for test maps f W C! A. We have to show: B is closed in A or, equivalently, in X. If g W D! X is a test map, then g1.A/ is closed in D and hence compact, since D is compact. The restriction of g yields a continuous map h W g1.A/! A. The set h1.B/ D g1.B/ is closed in g1.A/ and therefore in D, and this shows that B is closed in X. Let U be open in the k-space X. We write X as quotient q W Z! X of a sum Z of ch-spaces (see 7.9.3). Then q W q1.U /! U is a quotient map and q1.U / as the topological sum of locally compact Hausdorff spaces is a k-space. Therefore the quotient U is a k-space. The second assertion follows, if we take (7.9.1) into account. In general, a subspace of a k-space is not a k-
space (see (7.9.23)). Let X be a k-space and i W A X the inclusion. Then the map k.i/ W k.A/! X D k.X/ is continuous. The next proposition shows that k.i/ has in the category k- TOP the formal property of a subspace. (7.9.11) Proposition. A map h W Z! k.A/ from a k-space Z into k.A/ is continuous if and only if k.i/ ı h is continuous. Proof. If h is continuous then also is k.i/ıh. Conversely, let k.i/ıh be continuous. We have k.i/ D i ı A. Since i is the inclusion of a subspace, A ı h is continuous; (7.9.6) now shows that h is continuous. (7.9.12) Theorem. The product in TOP of a k-space X with a locally compact Hausdorff space Y is a k-space. Proof. By (7.9.8), a locally compact Hausdorff space is a k-space. We write X as quotient of Z, where Z is a sum of ch-spaces, see (7.9.3). Since the product of a quotient map with a locally compact space is again a quotient map, we see that X Y is a quotient of the locally compact Hausdorff space, hence k-space, Z Y, and therefore a k-space by (7.9.9). A product of k-spaces is not always a k-space (see (7.9.23)). Therefore one is looking for a categorical product in the category k- TOP. Let.Xj j j 2 J / be a j Xj its product in the category TOP, i.e., the ordinary family of k-spaces and topological product. We have a continuous map Q pj D k.prj / W k Q j Xj! k.Xj / D Xj : The following theorem is a special case of the fact that a right adjoint functor respects limits. (7.9.13) Theorem. in the category k- TOP. pj W k 7.9. Compactly Generated Spaces 191 Q j Xj! Xj j j 2 J is a product of.
Xj j j 2 J / Proof. We use (7.9.6) and the universal property of the topological product and obtain, in short-hand notation, for a k-space B the canonical bijection Q Q Q Q k- TOP B; k Xj D TOP B; Xj Š TOP.B; Xj / D k- TOP.B; Xj /; and this is the claim. In the case of two factors, we use the notation X k Y for the product in k- TOP just deﬁned. The next result shows that the wh-spaces are the formally hausdorff spaces in the category k- TOP. (7.9.14) Proposition. A k-space X is a wh-space if and only if the diagonal DX of the product X k X is closed. In order to verify that DX is closed, we have to Proof. Let X be a wh-space. show that for each test map f W K! X k X the pre-image f 1.DX / is closed. Let fj W K! X be the j -th component of f. Then Lj D fj.K/ is a ch-space, since X is a wh-space. Hence L D L1 [ L2 X is a ch-space. The relation f 1DX D f 1..L L/ \ DX / shows that this set is closed. Let DX be closed in X k X and f W K! X a test map. We have to show that f.K/ X is closed. Let g W L! X be another test map. Since X is a k-space, we have to show that g1f.K/ L is closed. We use the relation g1f.K/ D pr2..f g/1DX /: Since DX is closed, the pre-image under f g is closed and therefore also its image under pr2 as a compact set in a Hausdorff space. Recall the mapping space F.X; Y / with compact-open topology. (7.9.15) Theorem. Let X and Y be k-spaces, and let f W X k Y! Z be continuous. The adjoint map f ^ W X! kF.Y; Z/, which exists as a set map, is continuous. Proof. The map f ^ W X! kF.Y;
Z/ is continuous, if for each test map t W C! X the composition f ^ıt is continuous. We use f ^ıt D.f ı.t idY //^. Therefore it sufﬁces to assume that X is a ch-space. But then, by (7.9.12), X k Y D X Y and therefore f ^ W X! F.Y; Z/ is continuous and hence also f ^ W X! kF.Y; Z/, by (7.9.4). (7.9.16) Theorem. Let Y be a k-space. Then the evaluation eY;Z W kF.Y; Z/ k Y! Z;.f; y/ 7! f.y/ is continuous. 192 Chapter 7. Stable Homotopy. Duality Proof. Let t W C! kF.Y; Z/ k Y be a test map. We have to show the continuity of eY;Z ı t. Let t ^ 1 W C! F.Y; Z/ and t2 W C! Y be the continuous components of t. We show ﬁrst: The adjoint t1 W C Y! Z of t ^ 1 is continuous. By (2.4.3), this continuity is equivalent to the continuity of the second adjoint map t _ 1 W Y! F.C; Z/. In order to show its continuity, we compose with a test map s W D! Y. But t _ 1 is continuous. Moreover we have eY;Z ı t D t1 ı.id; t2/, and the right-hand side is continuous. 1 ı s D F.s; Z/ ı t ^ A combination of (7.9.15) and (7.9.16) now yields the universal property of the evaluation eY;Z for k-spaces: (7.9.17) Proposition. Let X and Y be k-spaces. The assignments f 7! f ^ and g 7! eY;Z ı.g k idY / D g are inverse bijections TOP.X k Y; Z/ Š TOP.X; kF.Y; Z// between these sets. (7.9.18) Theorem. Let X, Y and Z be k-spaces. Since eY;Z is continuous,
we have an induced set map W kF.X; kF.Y; Z//! kF.X k Y; Z/; f 7! eY;Z ı.f k idY / D f : The map is a homeomorphism. Proof. We use the commutative diagram kF.X; kF.Y; Z// k X k Y e1id kF.Y; Z/ k Y id id kF.X k Y; Z/ k X k Y e3 e2 Z with e1 D eX;kF.Y;Z/, e2 D eY;Z, and e3 D eXk Y;Z. Since e1 id and e2 are continuous, the universal property of e3 shows that is continuous; namely, using the notation from (7.9.17), we have e2 ı.e1 id/ D. The universal property of e1 provides us with a unique continuous map W kF.X k Y; Z/! kF.X; kF.Y; Z//; f 7! f ^; such that e1 ı. id.X// D e^ 3 W kF.X k Y; Z/ k X! kF.Y; Z/ is the adjoint of e3 with respect to the variable Y. One checks that and are inverse to each other, hence homeomorphisms. 3, where e^ (7.9.19) Theorem. Let X and Y be k-spaces, and f W X! X 0 and g W Y! Y 0 be quotient maps. Then is a quotient map. 7.9. Compactly Generated Spaces 193 Proof. It sufﬁces to treat the case g D id, since a composition of quotient maps is a quotient map. Using (7.9.18), the proof is now analogous to (2.4.6). (7.9.20) Proposition. Let f W X! Y be a quotient map and X a whk-space. Then Y is a whk-space if and only if R D f.x1; x/ j f.x1/ D f.x2/g is closed in X k X. Proof. The set R is the pre-image of DY under f f. Since f k f is a quotient map (7
.9.19), DY is closed if and only if R is closed. Now apply (7.9.9) and (7.9.14). (7.9.21) Proposition. Let Y and Z be k-spaces and assume that Z is a wh-space. Then the mapping space kF.Y; Z/ is a wh-space. In particular, if Y and Z are whk-spaces, then kF.Y; Z/ is a whk-space. Proof. Let f ^ W K! kF.Y; Z/ be a test map. We have to show that it has a closed image hence is k-closed. For this purpose let g^ W L! kF.Y; Z/ be another test map. It remains to show that the pre-image M of f ^.K/ under g^ is closed. We use the adjoint maps f W K Y! Z and g W L Y! Z. For y 2 Y let iy W K L!.K Y / k.L Y /,.k; l/ 7!.k; y; l; y/. Then. Since Z is a wh-space and therefore the M D pr2 diagonal DZ is closed by (7.9.14), we see that M is closed. y2Y..f g/iy/1DZ T Q V We now consider pointed spaces. Let.Xj j j 2 J / be a family of pointed k j Xj be its product in k- TOP. Let WJ Xj be the subset of the k-spaces. Let product of those points for which at least one component equals the base point. Q =WJ Xj. In the case that The smash product J D f1; : : : ; ng we denote this space by X1 ^k ^k Xn. A family of pointed maps V k k fj W fj W Xj! Yj induces a pointed map j Yj. Let X and Y be pointed k-spaces. Let F 0.X; Y / F.X; Y / be the subspace of pointed maps. We compose a pointed map f W X ^k Y! Z with the projections p W X k Y! X ^k Y. The adjoint.fp/^ W X! kF.Y; Z/ is continuous and has an image contained in kF 0
.Y; Z/. We obtain a continuous map X! kF 0.Y; Z/ which will be denoted by f ^. k j Xj is the quotient space V k j Xj! k j Xj V The evaluation eY;Z induces e0 Y;Z which makes the following diagram commu- tative: kF 0.Y; Z/ k X k.i/id kF.Y; Z/ k X p kF 0.Y; Z/ ^k X e0 Y;Z eY;Z Y. i is the inclusion and p the quotient map. The continuity of k.i/ and eY;Z implies the continuity of the pointed evaluation e0 Y;Z. In analogy to (7.9.18) one proves: 194 Chapter 7. Stable Homotopy. Duality (7.9.22) Theorem. Let X, Y and Z be pointed k-spaces. The assignment 0 W kF 0.X ^k Y; Z/! kF 0.X; kF 0.Y; Z/; f 7! f ^ is a homeomorphism. (7.9.23) Example. Let R=Z be obtained from R by identifying the subset Z to a point (so this is not the factor group!). We denote by p W R! R=Z the quotient map. (1) The product p id W R Q! R=Z Q of quotient maps is not a quotient map. (2) The product R=Z Q is not a k-space, but the factors are k-spaces (see (7.9.4)). (3) The product R=Z R is a k-space (see (7.9.9) and (7.9.12)), but the subspace R=Z Q is not a k-space by (2). If K R=Z is compact, then there exists l 2 N such that K pŒl; l. p Let.rn j n 2 N/ be a strictly decreasing sequence of rational numbers with R Q is saturated with 2. The set F D j n; m 2 N ˚ 2n ; rn limit m respect to p id and closed in R Q. m C 1 p The set G D.pid/.F / is not closed in R=ZQ. Note that z D.p.0
/; 0/ 62 G; but we show that z 2 xG. Let U be a neighbourhood of z. Then there exists a neighbourhood V of p.0/ in R=Z and " > 0 such that V. "; "Œ \Q/ U. Choose m 2 N such that m1 2 < 21". The set p1.V / is then a neighbourhood of m in R, since m 2 p1p.0/ p1.V /. Hence there exists ı > 0 such 2n < ı and that m / 2 V. "; "Œ \Q/ U holds, rn m C rn because mC 1 2 D ". We see that U \ G 6D ;. This ﬁnishes the proof that z 2 xG. m ı; m C ıŒ p1.V /. Now choose n 2 N such that 1 p 2n 2 mı; mCıŒ p1.V / and 0 < rn 2. Then.p id/.m C 1 m < " 2 < m " 2n ; rn 2 C " m D p p 2 2 We now see that p id is not a quotient map, since there exists a saturated closed set F with non-closed image G. The space R=Z Q is not a k-space. Let s W K! R=Z Q be an arbitrary test map. We show that s1.G/ is closed in K although G is not closed (this could not occur in a k-space). The two projections pri s.K/ are compact and Hausdorff. Hence there exists l 2 N such that pr1 s.K/ pŒl; l. The inclusion s.K/ pr1 s.K/ pr2 s.K/ pŒl l pr2 s.K/ then shows that we have s1.G/ D s1.G \ pŒl; l pr2 s.K//. But the set G \ pŒl; l pr2 s.K/ is ﬁnite: By construction, F is a closed discrete subspace of R Q; moreover, F \ Œl; l pr2 s.K/ is ﬁnite as a closed discrete subspace of the compact space Œl; l pr2 s.K/; therefore also.p id/.
F \ Œl; l pr2 s.K// D G \ pŒl; l pr2 s.K/ 7.9. Compactly Generated Spaces 195 is ﬁnite. A ﬁnite set in a Hausdorff space is closed, and therefore s1.G/ as preÞ image of a closed set is closed itself. (7.9.24) Example. It is stated already in [155, p. 336] that.Q ^ Q/ ^ N0 and Q ^.Q ^ N0/ are not homeomorphic. In [128, p. 26 ] it is proved that the canonical Þ continuous bijection from the ﬁrst to the second is not a homeomorphism. Problems S ` 1. A space is a k-space if and only if it is a quotient of a locally compact Hausdorff space. 2. Let X1 X2, let Xj be a whk-space and let Xj Xj C1 be closed. Then X D j Xj, with colimit topology, is a whk-space. If the Xi are k-spaces, then X is a i Xi. If the Xi are wh-spaces, hence T1-spaces, k-space, being a quotient of the k-space then each test map f W K! X has an image which is contained in some Xi and therefore closed. If each inclusion is Xi XiC1 closed, the image is also closed in X and therefore X is a wh-space. 3. Let X and Y be k-spaces. Passage to adjoint maps induces bijections of homotopy sets ŒX k Y; Z Š ŒX; kF.Y; Z/ and ŒX ^ Y; Z0 Š ŒX; kF 0.Y; Z/0. 4. Let.Xj j j 2 J / be a family of k-spaces. Then the topological sum k-space. The product in k-TOP is compatible with sums. 5. Let a pushout of topological spaces with closed j W A X be given. j 2J Xj is Let X and B be whk-spaces. Then Y is a whk-space. Chapter 8 Cell Complexes The success of algebraic topology is largely due to the fact that one
can describe spaces of interest by discrete (or even ﬁnite) combinatorial data. Purely combinatorial objects are simplicial complexes. Given such a complex, one deﬁnes from its data by simple linear algebra the homology groups. It is then a remarkable fact that these groups are independent of the combinatorial description and even homotopy invariant. Simplicial complexes are a very rigid structure. A weakening of this structure is given by the cell complexes (CW-complexes in the sense of J. H. C. Whitehead). They are more ﬂexible and better adapted to homotopy theory. An n-cell in a space is a subset which is homeomorphic to the standard n-cell E n D fx 2 Rn j kxk < 1g. A cell complex is a decomposition of a space into cells. In order that one obtains something interesting, one has to add conditions about the closure of the cells, and one has to relate the topology of the space to the topology of the closed cells. A ﬁnite cell complex is easily deﬁned: a Hausdorff space X which is the union of a ﬁnite number of cells. It e is an n-cell of this decomposition, then it is required that there exists a continuous map'W Dn! X which induces a homeomorphism E n! e and sends S n1 into the union of the cells up to dimension n 1. P From these data one obtains already an interesting invariant of X, the so-called Euler characteristic. Let n.i/ denote the number of i-cells. Deﬁne the combinatorial i0.1/i n.i/. It is a nonEuler characteristic to be the alternating sum.X / D trivial fact that h-equivalent ﬁnite complexes have the same Euler characteristic. The origin of the notion is the famous result of Leonhard Euler ( 1752) that for a sphere S 2 each polyhedral decomposition yields the value n.0/ n.1/ C n.2/ D 2. In this chapter we present some point-set topology and elementary homotopy theory of cell complexes. Then we demonstrate the use of cell complexes in the construction of spaces with speciﬁc properties. In particular we construct so-called Eilenberg–
Mac Lane spaces K.; n/. They have a single non-vanishing homotopy group n.K.; n// Š (here can be an arbitrary abelian group). Eilenberg–Mac Lane spaces can be used as building blocks for general homotopy types (Postnikov systems). They also yield a homotopical deﬁnition of cohomology (and homology) groups: The homotopy set ŒX; K.; n/ carries a natural structure of an abelian group and is known to be a version of a cohomology group denoted H n.XI /. 8.1. Simplicial Complexes 197 8.1 Simplicial Complexes Simplicial complexes are the objects of combinatorial topology. A simplicial complex K D.E; S/ consists of a set E of vertices and a set S of ﬁnite non-empty subsets of E. A set s 2 S with q C 1 elements is called a q-simplex of K. We require the following axioms: (1) feg 2S for each e 2 E. (2) If t 2 S and s t is non-empty, then s 2 S. If s 2 S is a q-simplex, then q is called the dimension of s. If t s, then t is a face of s. A 1-simplex of K is also called an edge of K. The 0-simplices of K correspond to the elements of E; a 0-simplex is called a vertex. A simplex is determined by its 0-faces. A simplicial complex is n-dimensional, if it contains at least one n-simplex but no.n C 1/-simplices. A subcomplex L of K consists of a set of simplices of K which contains with s also the faces of s. A 1-dimensional complex is a graph. A complex K D.E; S/ is ﬁnite if E is ﬁnite and locally ﬁnite if each vertex is contained in a ﬁnite number of simplices. The n-skeleton Kn D.E; S n/ of K D.E; S / is the subcomplex with S n D fs 2 S j dim s ng. (8.1.1) Example. Let U D.Uj j j 2 J / be
a covering of a set X by nonempty sets Uj. For a ﬁnite non-empty set E J let UE D j 2E Uj and let E.J / D fE J j UE 6D ;g. Then.J; E.J // is a simplicial complex, called the Þ nerve N.U/ of the covering U. T (8.1.2) Example. Let P be a set with a partial ordering. The simplicial complex.P; SP / associated to a partially ordered set has as simplices the totally ordered Þ ﬁnite subsets of P. (8.1.3) Example. Let K D.E; S/ be a simplicial complex. Deﬁne a partial order on S by s t, s t. The simplicial complex K0 associated to this ordered set Þ is called the barycentric subdivision of K. Let K D.E; S/ be a simplicial complex. We denote by jKj the set of functions ˛ W E! I such that P (1) fe 2 E j ˛.e/ > 0g is a simplex of K. (2) e2E ˛.e/ D 1. We regard jKj as a subset of the product I E. Let jKjp denote this set with the subspace topology of the product topology. We have a metric d on jKj deﬁned by d.˛; ˇ/ D P e2E.˛.e/ ˇ.e//2 1 2 : We denote jKj with this metric topology by jKjm. Each vertex e 2 E gives us a continuous map e W jKjm! I, ˛ 7! ˛.e/. Therefore the identity jKjm! jKjp is 198 Chapter 8. Cell Complexes continuous. We leave it as an exercise to show that this map is actually a homeomorphism. The numbers.˛.e/ j e 2 E/ are the barycentric coordinates of ˛. We deﬁne a further topology on jKj. For s 2 S let.s/ be the standard simplex f.te/ 2 jKj jt e D 0 for e … sg. Then jKj is the union
of the.s/, and we write jKjc for jKj with the quotient topology deﬁned by the canonical map ` s2S.s/! jKj. The identity jKjc! jKjp is continuous but not, in general, a homeomorphism. The next proposition will be proved in the more general context of simplicial diagrams. (8.1.4) Proposition. jKjc! jKjp is a homotopy equivalence. In the sequel we write jKj D jKjc and call this space the geometric realization 6D 0 ) e 2 sg and of K. We deﬁne jsj jKj as jsj D f˛ 2 jKj j˛.e/ call this set a closed simplex of jKj. For each simplex s of K the open simplex hsi jKj is the subspace hsi D f˛ 2 jKj j˛.e/ 6D 0, e 2 sg. The complement jsj n hsi D@js j is the combinatorial boundary of jsj; it is the geometric realization of the subcomplex which consists of the proper faces of s. The set jKj is the disjoint union of the hsi; s 2 S. Let jKjn be the union of the.s/ with dim s n. (8.1.5) Proposition. The space jKj is the colimit of the jKjn. The equality jKnj D jKjn holds. The canonical diagram ` s;dim sDn @.s/ ` s;dim sDn.s/ jKjn1 jKjn is a pushout. A homeomorphism t W jKj!X is called a triangulation of X. The triangulation of surfaces was proved by Radó [161], the triangulation of 3-dimensional manifolds by Moise (see [141] for references and proofs; [197, 7.5.1]). Differentiable manifolds can be triangulated, and the triangulation can be chosen in such a way that it is on each simplex a smooth embedding ([193]; [143]). Since jKjm is separated and id W jKj! jKjm continuous, jKj is separated. For ﬁnite K
the identity jKj! jKjm is a homeomorphism. For each vertex e 2 E the set St.e/ D f˛ 2 jKj j˛.e/ 6D 0g is called the star of e. Since ˛ 7! ˛.e/ is continuous, the set St.e/ is open in jKjd and therefore also in jKj. If we identify e with the function ˛.e/ D 1; ˛.e0/ D 0 for e 6D e0, then St.e/ is an open neighbourhood of e. Points e0; : : : ; ek of Rn are afﬁnely independent, if the relations †i ei D 0 and †i D 0 imply that each i D 0. If e0; : : : ; ek are afﬁnely independent, then spanned by e0; : : : ; ek is the convex the simplex hull of this set and homeomorphic to the k-dimensional standard simplex. ˚ P k iD0 i ei j i 0; †i D 1 8.2. Whitehead Complexes 199 Let K D.E; S/ be a simplicial complex and.xe j e 2 E/ a family of points in Rn. Consider the continuous map f W jKj!R n; ˛ 7! P e2E ˛.e/xe: If f is an embedding, we call the image of f a simplicial polyhedron in Rn of type K, and f.jKj/ is a realization of K as a polyhedron in Rn. Standard tools for the application of simplicial complexes in algebraic topology are subdivision and simplicial approximation [67, p. 124]. Problems T Lj are subcomplexes and the relations 1. id W jKjm! jKjp is a homeomorphism. 2. Let K D.N0; S/ be the simplicial complex where S consists of all ﬁnite subsets of N0. The canonical map jKjc! jKjp is not a homeomorphism. 3. Let L be a subcomplex of K. We can identify jLj with a subset of jKj, and jLj carries then the subspace topology of j
Kj. If.Lj j j 2 J / is a family of subcomplex of K, then Lj Lj j jLj j D j and hold. 4. Let K be a simplicial complex. Then the following assertions are equivalent: (1) K is locally ﬁnite. (2) jKj is locally compact. (3) The identity jKj! jKjd is a homeomorphism. (4) jKj is metrizable. (5) Each point of jKj has a countable neighbourhood basis. (See [44, p. 65].) 5. Let K be a countable, locally ﬁnite simplicial complex of dimension at most n. Then K has a realization as a polyhedron in R2nC1. (See [44, p. 66].) jLj j D j Lj j and S T S S T 8.2 Whitehead Complexes We use the standard subsets of Euclidean spaces S n1; Dn; En D Dn n S n1,.n 1/. We set S 1 D ; and let D0 be a point, hence E 0 D D0. A k-dimensional cell (a k-cell) in a space X is a subset e which is, in its subspace topology, homeomorphic to Ek. A point is always a 0-cell. A Whitehead complex is a space X together with a decomposition into cells.e j 2 ƒ/ such that: (W1) X is a Hausdorff space. (W2) For each n-cell e there exists a characteristic map ˆ W Dn D Dn! X which induces a homeomorphism E n! e and sends S n1 into the union X n1 of the cells up to dimension n 1. (W3) The closure xe of each cell e intersects only a ﬁnite number of cells. (W4) X carries the colimit topology with respect to the family.xe j 2 ƒ/. A subset A of a Whitehead complex is a subcomplex if it is a union of cells and the closure of each cell in A is contained in A. We will see that a subcomplex together with its cells is itself a Whitehead complex. From the deﬁnition of a subcomplex 200 Chapter 8. Cell
Complexes we see that intersections and unions of subcomplexes are again subcomplexes. Therefore there exists a smallest subcomplex X.L/ which contains a given set L. The decomposition of a Hausdorff space into its points always satisﬁes (W1)– (W3). We see that (W4) is an important condition. Condition (W3) is also called (C), for closure ﬁnite. Condition (W4) is called (W), for weak topology. This is the origin for the name CW-complex. In the next section we consider these complexes from a different view-point and introduce the notion of a CW-complex. (8.2.1) Lemma. Let ˆ W Dn! X be a continuous map into a Hausdorff space. Let e D ˆ.En/. Then ˆ.Dn/ D xe. In particular xe is compact. Proof. ˆ.Dn/ is a compact subset of a Hausdorff space and therefore closed. This yields xe D ˆ.En/ ˆ.Dn/ D ˆ.Dn/ D ˆ. SEn/ ˆ.En/ D xe: (8.2.2) Example. Suppose X has a cell decomposition into a ﬁnite number of cells such that properties (W1) and (W2) hold. Then X is a ﬁnite union of closures xe of cells and therefore compact by (8.2.1). Properties (W3) and (W4) are satisﬁed and Þ X is a Whitehead complex. (8.2.3) Examples. The sphere S n has the structure of a Whitehead complex with a single 0-cell and a single n-cell. The map p ˆ W Dn! S n; x 7!.2 1 kxk2 x; 2kxk2 1/ sends S n1 to the 0-cell enC1 D.0; : : : ; 0; 1/ and induces a homeomorphism of E n with S n X fenC1g, hence is a characteristic map for the n-cell. From this cell decomposition we obtain a cell decomposition of DnC1 by adding another.n C 1/-
cell E nC1 with characteristic map the identity. Another cell-composition of S n has two j -cells for each j 2 f0; : : : ; ng and ˙ D f.xi / 2 S n j ˙xnC1 0g with intersection 1 kxk2/.Þ is obtained inductively from Dn S n1 D S n1 0. A characteristic map is Dn! Dn p ˙, x 7!.x; ˙ (8.2.4) Proposition. Let X be a Whitehead complex. (1) A compact set K in X meets only a ﬁnite number of cells. (2) A subcomplex which consists of a ﬁnite number of cells is compact and closed in X. (3) X.e/ D X.xe/ is for each cell e a ﬁnite subcomplex. (4) A compact subset of a Whitehead complex is contained in a ﬁnite subcomplex. (5) X carries the colimit topology with respect to the ﬁnite subcomplexes. (6) A subcomplex A is closed in X. Proof. (1) Let E be the set of cells which meet K. For each e 2 E we choose a point xe 2 K \ e and set Z D fxe j e 2 Eg. Let Y Z be any subset. For each cell f of X the closure xf is contained in the union of a ﬁnite number of cells. Thus 8.2. Whitehead Complexes 201 Y \ xf is a ﬁnite set, hence closed in xf since xf is a Hausdorff space. The condition (W4) now says that Y is closed in X and hence in Z. This tells us that Z carries the discrete topology and is closed in X. A discrete closed set in a compact space is ﬁnite. (2) Let A D e1[ [er be a ﬁnite union of cells ej. Then xA D xe1[ [xer A, by deﬁnition of a subcomplex. By (8.2.1), A D xA is compact and closed. (3) Induction over dim.e/. If e is a 0-cell, then e is a point and closed,
hence a subcomplex and X.e/ D X.xe/ D e. Suppose X.f / is ﬁnite for each cell f with dim.f / < n. Let e be an n-cell with characteristic map ˆ. The set ˆ.S n1/ is contained in the union of cells of dimension at most n 1, hence is contained in xe X e. Then xe X e D ˆ.S n1/ is compact, hence contained in a ﬁnite number of cells e1; : : : ; ek, by (1), which are contained in X n1, by (W2). By induction hypothesis, the set C D e [ X.e1/ [ [ X.ek/ is a ﬁnite subcomplex which contains e and hence X.e/. Therefore X.e/ is ﬁnite. Since X.e/ is closed, by (2), we have xe X.e/ and X.xe/ X.e/. (4) This is a consequence of (1) and (3). (5) We show: A X is closed if and only if for each ﬁnite subcomplex Y the intersection A \ Y is closed in Y. Suppose the condition is satisﬁed, and let f be an arbitrary cell. Then A\X.f / is closed in X.f /, hence, by (2) and (3), closed in X; therefore A \ xf D A \ X.f / \ xf is closed in X and in xf, hence closed in X by condition (W4). (6) If Y is a ﬁnite subcomplex, then A \ Y is a ﬁnite subcomplex, hence closed. By (5), A is closed. (8.2.5) Proposition. A subcomplex Y of a Whitehead complex X is a Whitehead complex. Proof. Let e be a cell in Y and ˆ W Dn! X a characteristic map. Then ˆ.Dn/ D xe Y, since Y is closed. Hence ˆ can be taken as a characteristic map for Y. It remains to verify condition (W4). Let L Y and suppose L \ xe is closed in xe for each cell e in Y. We have to show that L is
closed in Y. We show that L is closed in X. Let f be a cell of X. By (W3), xf is contained in a ﬁnite union e1 [ [ ek of cells. Let e1; : : : ; ej be those which are contained in Y. Then xf \ Y e1 [ [ ej xe1 [ [ xej Y since Y is a subcomplex. Hence xf \ Y D. xf \ xe1/ [ [. xf \ xej /; xf \ L D xf \ L \ Y D S j kD1. xf \ xek \ L/: By assumption, xek \ L is closed in xek; hence xf \ xek \ L is closed in xf ; therefore xf \ L is a ﬁnite union of sets which are closed in X. 202 Chapter 8. Cell Complexes (8.2.6) Proposition. Let X be a Whitehead complex. Then: (1) X carries the colimit topology with respect to the family.X n j n 2 N0/. (2) Let.e j 2 ƒ.n// be the family of n-cells of X with characteristic maps ˆ W Dn! X n and restrictions'W S n1 ` 'Dh'i! X n1. Then S n1 i ` Dn ˆDh ˆ i X n1 \ X n is a pushout in TOP. (X 1 D ;.) Proof. (1) Suppose A \ X n is closed in X n for each n. Then for each n-cell e of X the set A \ xe D A \ xe \ X n is closed in xe. By (W4), A is closed in X. (2) The diagram is a pushout of sets. Give X n the pushout topology and denote this space by Z. By construction, the identity W Z! X n is continuous. We show that is also closed. Let V Z be closed. By deﬁnition of the pushout topology this means: (i) V \ X n1 is closed in X n1. is closed in Dn (ii) ˆ1.V / \ Dn, hence also compact. We conclude that ˆ.ˆ1.V /
\ Dn / D V \ ˆ.Dn / D V \ xe is closed in xe, being a continuous image of a compact space in a Hausdorff space. From (i) and (ii) we therefore conclude that for each cell e of X n the set V \ xe is closed in xe. Since X n is a Whitehead complex, V is closed in X n. (8.2.7) Proposition. Let X be a Whitehead complex, pointed by a 0-cell. The inclusions of the ﬁnite pointed subcomplexes F X induce a canonical map colimF k.F; /! k.X; /. This map is an isomorphism. Recall from Section 7.9 the notion of a k-space and the k-space k.X/ obtained from a space X. (8.2.8) Proposition. Let X have a cell decomposition such that (W1)–(W3) hold and such that each compact set is contained in a ﬁnite number of cells. Then k.X/ is a Whitehead complex with respect to the given cell decomposition and the same characteristic maps. Moreover, X is a Whitehead complex if and only if k.X/ D X. Proof. Let ˆ W Dn! X be a characteristic map for the cell e. Since xe is compact it has the same topology in k.X /. Hence ˆ W Dn! k.X/ is continuous. Since ˆ is a quotient map and ˆ1.e/ D E n, we see that e has the same topology in k.X/ and X. Thus e is a cell in k.X/ with characteristic map ˆ. Let A \ xe be closed in xe for each cell e. Let K k.X/ be compact. By hypothesis, K is contained in a ﬁnite number of cells, say K e1 [ [ ek. Then A \ K D..A \ xe1/ [ [.A \ xek// \ K is closed in K. Hence A is k-closed. 8.3. CW-Complexes 203 Let X and Y be Whitehead complexes and e X, f Y cells. Then e f X Y is a cell. From characteristic maps ˆ W Dm! X,
‰ W Dn! Y for e; f we obtain ˆ ‰ W Dm Dn! X Y, and this can be considered as a characteristic map for e f. For this purpose use a homeomorphism.DmCn; S mCn1/!.Dm Dn; Dm S n1 [ S m1 Dn/: With this cell structure, X Y satisﬁes conditions (W1)–(W3) in the deﬁnition of a Whitehead complex. In general, property (W4) may not hold. In this case one re-topologizes X Y such that the compact subsets do not change. The space X k Y D k.X Y / is then a Whitehead complex (see (8.2.8)). Problems 1. R carries the structure of a Whitehead complex with 0-cells fng, n 2 Z and 1-cells n; n C 1Œ, n 2 Z. There is an analogous Whitehead complex structure W.ı/ on Rn with 0-cells the set of points ı.k1; : : : ; kn/, kj 2 Z, ı > 0 ﬁxed and the associated ı-cubes. Thus, given a compact set K Rn and a neighbourhood U of K, there exists another neighbourhood L of K contained in U such that L is a subcomplex of the complex W.ı/. In this sense, compact subsets can be approximated by ﬁnite complexes. 2. The geometric realization of a simplicial complex is a Whitehead complex. 8.3 CW-Complexes We now use (8.2.6) as a starting point for another deﬁnition of a cell complex. Let.X; A/ be a pair of spaces. We say, X is obtained from A by attaching an n-cell, if there exists a pushout S n1 \ Dn'ˆ A \ X. Then A is closed in X and X n A is homeomorphic to E n via ˆ. We call X n A an n-cell in X,'its attaching map and ˆ its characteristic map. (8.3.1) Proposition. Let a commutative diagram with closed embeddings j; J be given: A j X f F Y J Z. Suppose F
induces a bijection X X A! Z X Y. Then the diagram is a pushout, provided that (1) F.X/ Z is closed; (2) F W X! F.X/ is a quotient map. Condition.2/ holds if X is compact and Z Hausdorff. 204 Chapter 8. Cell Complexes Proof. Let g W X! U and h W Y! U be given such that gj D hf. The diagram is a set-theoretical pushout. Therefore there exists a unique set map'W Z! U with 'F D g, 'J D h. Since J is a closed embedding, 'jJ.Y / is continuous. Since F is a quotient map, 'jF.X/ is continuous. Thus'is continuous, since F.X/ and J.Y / are closed sets which cover Z. (8.3.2) Note. Let X be a Hausdorff space and A a closed subset. Suppose there exists a continuous map ˆ W Dn! X which induces a homeomorphism ˆ W En! X n A. Then X is obtained from A by attaching an n-cell. Proof. We show ˆ.S n1/ A. Suppose there exists s 2 S n1 with ˆ.s/ 2 X nA. Then there exists a unique t 2 En with ˆ.s/ D ˆ.t/. Let V En, W Dn be disjoint open neighbourhoods of t; s. Then ˆ.V / X n A is open in X, since ˆ W E n! X n A is a homeomorphism and A is closed in X. Since ˆ is continuous, there exists an open neighbourhood W1 W of s with ˆ.W1/ ˆ.V /. This contradicts the injectivity of ˆjE n. Thus ˆ provides us with a map'W S n1! A. We now use (8.3.1). (8.3.3) Example. The projective space RP n is obtained from RP n1 by attaching an n-cell. The projective space CP n is obtained from CP n1 by attaching a 2ncell. We recall that CP n1 is obtained from S 2n1 by the equivalence relation.z1; : : :
; zn/.z1; : : : ; zn/, 2 S 1, or from Cn n 0 by z z, 2 C. The class of z is denoted Œz1; : : : ; zn. A characteristic map ˆ W D2n! CP n is x 7! Œx; 1 kxk2. p The space RP n1 is obtained from S n1 by the relation z z, or from Rn n 0 by z z, 2 R. A characteristic map ˆ W Dn! RP n is given by the same Þ formula as in the complex case. We can also attach several n-cells simultaneously. We say X is obtained from A by attaching n-cells if there exists a pushout ` j 2J S n1 j'` \ j 2J Dn j ˆ A \ X. The index j just enumerates different copies of the same space. Again, A is then closed in X and ˆ induces a homeomorphism of j with X n A. Therefore X n A is a union of components and each component is an n-cell. (By invariance of dimension, the integer n is determined by X n A.) We allow J D ;; in that case A D X. We write ˆj D ˆjDn j and 'j D 'jS n1 and call ˆj the characteristic map of the n-cell ˆ.En j / and 'j its attaching map. j En ` 8.3. CW-Complexes 205'pr S n J Let us give another interpretation: X D X.'/ is the double mapping cylinder of J! A where J is a discrete set. From this setting we see: If'is replaced by a homotopic map, then X.'/ and X. / are h-equivalent under A. Let f W A! Y be a given map. Assume that X is obtained from A by attaching j! A. From the pushout deﬁnition of j S n1 ` n-cells via attaching maps h'j i W the attaching process we obtain: (8.3.4) Note. There exists an extension F W X! Y of f if and only if the maps f 'j are null homotopic. We view a null homotopy of f 'j as an extension to Dn j
. Then the extensions F correspond to the set of null homotopies of the f 'j. In view of this note we call the homotopy classes Œf 'j the obstructions to extending f. Let A be a subspace of X. A CW-decomposition of.X; A/ consists of a sequence of subspaces A D X 1 X 0 X 1 X such that: (1) X D [n0X n. (2) For each n 0, the space X n is obtained from X n1 by attaching n-cells. (3) X carries the colimit topology with respect to the family.X n/. Xk is a subspace of the colimit X of a sequence Xj Xj C1. If the inclusions are closed, then Xk is closed in X. This is an immediate consequence of the deﬁnition of the colimit topology. A pair.X; A/ together with a CW-decomposition.X n j n 1/ is called a relative CW-complex. In the case A D ; we call X a CW-complex. The space X n is the n-skeleton of.X; A/ and.X n j n 1/ is the skeleton ﬁltration. The cells of X n n X n1 are the n-cells of.X; A/. We say,.X; A/ is ﬁnite (countable etc.) if X n A consists of a ﬁnite (countable etc.) number of cells. If X D X n, X ¤ X n1 we denote by n D dim.X; A/ the cellular dimension of.X; A/. If A D ;, then A is suppressed in the notation. We call X a CW-space if there exists some cellular decomposition X 0 X 1 of X. Let X be a Whitehead complex. From (8.2.6) we obtain a CW-decomposition of X. The converse also holds: From a CW-decomposition we obtain a decomposition into cells and characteristic maps; it remains to verify that X is a Hausdorff space and carries the colimit topology with respect to the closures of cells (see (8.3.8)). In the context of CW-complexes.X; A/, the symbol X n usually denotes the n
-skeleton and not the n-fold Cartesian product. (8.3.5) Note. If.X; A/ is a relative CW-complex, then also.X; X n/ and.X n; A/ are relative CW-complexes, with the obvious skeleton-ﬁltration inherited from.X n j n 1/. (8.3.6) Example. From (8.3.3) we obtain cellular decompositions of CP n and RP n. The union of the sequence RP n RnC1 deﬁnes the inﬁnite projective 206 Chapter 8. Cell Complexes space RP 1 as a CW-complex. It has a single n-cell for each n 0. Similarly, we obtain CP 1 with a single cell in each even dimension. Þ (8.3.7) Example. The sphere S n has a CW-decomposition with a single 0-cell and a single n-cell, and another CW-composition with two j -cells for each j 2 f0; : : : ; ng, see (8.2.3). The quotient map S n! RP n sends each cell of the latter homeomorphically onto a cell of RP n in the decomposition (8.3.6). We can also form the colimit S 1 of S n S nC1, a CW-complex with two cells in each Þ dimension. The general topology of adjunction spaces and colimit topologies gives us the next results. (8.3.8) Proposition. Let.X; A/ be a relative CW-complex. If A is a T1-space, then X is a T1-space and a compact subset of X meets only a ﬁnite number of cells. If A is a Hausdorff space, then X is a Hausdorff space. If A is normal, then X is normal. If A is a Hausdorff space, then X carries the colimit topology with respect to the family which consists of A and the closures of cells. Proof. We only verify the last statement. Let C be a subset of X and suppose A\C in closed in A and A \ xe closed in xe for each cell e. We show inductively, that C \ X n is closed in X n. This holds for n D 1 by
assumption. The space X n is a quotient of Zn D X n1 C j! xej is a quotient map, since X is Hausdorff. From the assumptions we see that X n \ C has a closed pre-image in Zn. j. Each characteristic map ˆj W Dn Dn ` The considerations so far show that a CW-complex is a Whitehead complex. (8.3.9) Proposition. Let.X; A/ be a relative CW-complex. Then A X is a coﬁbration. ` ` Dn S n1 j! j is a coﬁbration. Hence X n1 X n is Proof. We know that an induced coﬁbration. Therefore the compositions X n X nCk are coﬁbrations. Given f W X! Z and a homotopy h1 W X 1 I! Z of f jX 1, we can extend this inductively to homotopies hn W X nI! Z such that hnC1jX nI D hn. Since X I is the colimit of the X n I, the hn combine to a homotopy h W X I! Z. Problems j.Dn=S n1/j Š X=A. 1. The attaching map for the n-cells yields a homeomorphism 2. Let.X; A/ and.Y; B/ be relative CW-complexes. Consider X Y with the closed subspaces.X Y /n D nC1 iD1 X i Y ni ; n 1: S W 8.4. Weak Homotopy Equivalences 207 In favorable cases, the ﬁltration..X Y /n j n 1/ is a CW-decomposition of the pair.X Y; A B/. Let Y be locally compact. Then.X Y /n is obtained from.X Y /n1 by attaching ` ` ; S n1 n-cells. 3. Let.X; A/ be a relative CW-complex and let C A. Then.X=C; A=C / is a relative CW-complex with CW-decomposition.X n=C /. Moreover, X=A is a CW-complex. 4. Let A X be a subcomplex. Then X=A is a CW-complex.
5. Let A and B be subcomplexes of X. Then A=.A \ B/ is a subcomplex of X=B. 6. Let A be a subcomplex of B and Y another CW-complex. Then A ^k Y is a subcomplex of B ^k Y. 7. Let A be a CW-complex. Suppose X is obtained from A by attaching n-cells via attaching S n1 j! An1. Then X is a CW-complex with CW-decomposition X j D Aj maps'W for j < n and X j D Aj [.X X A/ for j n, and A is a subcomplex of X. 8. Let '0; '1 W j! A be homotopic attaching maps. The spaces X.0/; X.1/ which are obtained by attaching n-cells with '0; '1 are h-equivalent under A. (Homotopy theorem for coﬁbrations.) 9. Let X be a pointed CW-complex with base point a 0-cell. Then the cone CX and the suspension †X are CW-complexes. (In statements of this type the reader is asked to ﬁnd a canonical cell decomposition induced from the initial data.) 10. Let.Xj j j 2 J / be a family of pointed CW-complexes with base point a 0-cell. Then W j 2J Xj has the structure of a CW-complex such that the summands are subcomplexes. 11. Let p W E! B be a Serre ﬁbration and.X; A/ a CW-pair. Then each homotopy h W X I! B has a lifting along p with given initial condition on X 0 [ A I. 12. Suppose X is obtained from A by attaching n-cells. Let p W E! X be a covering and E0 D p1.A/. Then E is obtained from E 0 by attaching n-cells. 13. Let X be a CW-complex with n-skeleton X n and p W E! X a covering. Then E is a CW-complex with n-skeleton En D p1.B n/ such that p maps the cells of E homeomorphically to cells of X. An automorphism of p maps cells of E homeomorphically to cells. 14. Each neighbourhood U of a point x of a CW-complex
contains a neighbourhood V which is pointed contractible to x. A connected CW-complex has a universal covering. The universal covering has a cell decomposition such that its automorphism group permutes the cells freely. 15. Let X and Y be countable CW-complexes. Then X Y is a CW-complex in the product topology. 8.4 Weak Homotopy Equivalences We now study the notion of an n-connected map and of a weak homotopy equivalence in the context of CW-complexes. (8.4.1) Proposition. Let.Y; B/ be n-connected. Then a map f W.X; A/!.Y; B/ from a relative CW-complex.X; A/ of dimension dim.X; A/ n is homotopic 208 Chapter 8. Cell Complexes relative to A to a map into B. In the case that dim.X; A/ < n the homotopy class of X! B is unique relative to A. Proof. Induction over the skeleton ﬁltration. Suppose X is obtained from A by attaching q-cells via'W k! A, q n. Consider a commutative diagram k S q1 ` ` S q1 k'` Dq k A i\ f B j\ ˆ X F Y. Since.Y; B/ is n-connected, F ˆ is homotopic relative to to a map into B. Since the left square is a pushout, we obtain a homotopy of F from a pair of homotopies of F ˆ and F i which coincide on. Since we have homotopies S q1, we can use on A the constant homotopy. Altogether we of F ˆ relative to k obtain a homotopy of F relative to A to a map into B. S q1 k ` ` ` S q1 k For an arbitrary.X; A/ with dim.X; A/ n we apply this argument inductively. Suppose we have a homotopy of f relative to A to a map g which sends X k into B. By the argument just given we obtain a homotopy of gjX kC1 relative to X k which sends X kC1 into B. Since X kC1 X is a coﬁbration, we extend this homotopy to X. In the case that
n D 1, we have to concatenate an inﬁnite number of homotopies. We use the ﬁrst homotopy on Œ0; 1=2 the second on Œ1=2; 3=4 and so on. (Compare the proof of (8.5.4).) Suppose dim.X; A/ < n. Let F0; F1 W X! B be homotopic relative to A to f. We obtain from such homotopies a map.X I; X @I [ A I /!.Y; B/ which is the constant homotopy on A. We apply the previous argument to the pair.X I; X @I [ A I / of dimension n and see that the homotopy class of the deformation X! B of f is unique relative to A. (8.4.2) Theorem. Let h W B! Y be n-connected, n 0. Then h W ŒX; B! ŒX; Y is bijective (surjective) if X is a CW-complex with dim X < n (dim X n). If h W B! Y is pointed, then h W ŒX; B0! ŒX; Y 0 is injective (surjective) in the same range. Proof. By use of mapping cylinders we can assume that h is an inclusion. The surjectivity follows if we apply (8.4.1) to the pair.X; ;/. The injectivity follows, if we apply it to the pair.X I; X @I /. In the pointed case we deform.X; /!.Y; B/ rel fg to obtain surjectivity, and for the proof of injectivity we apply (8.4.1) to the pair.X I; X @I [ I /. (8.4.3) Theorem. Let f W Y! Z be a map between CW-complexes. (1) f is a homotopy equivalence, if and only if for each b 2 Y and each q 0 the induced map f W q.Y; b/! q.Z; f.b// is bijective. 8.4. Weak Homotopy Equivalences 209 (2) Suppose dim Y k, dim Z k. Then f is a homotopy equivalence if f
is bijective for q k. Proof. (1) If f is always bijective, then f is a weak equivalence, hence the induced map f W ŒX; Y! ŒX; Z is bijective for all CW-complexes X (see (8.4.2)). By category theory, f represents an isomorphism in h-TOP: Take X D Z; then there exists g W Z! Y such that fg'id.Z/. Then g is always bijective. Hence g also has a right homotopy inverse. (2) f W ŒZ; Y! ŒZ; Z is surjective, since f is k-connected (see (8.4.2)). Hence there exists g W Z! Y such that fg'id.Z/. Then g W q.Z/! q.Y / is bijective for q k, since fg D id and f is bijective. Hence there exists h W Y! Z with gh'id.Y /. Thus g has a left and a right h-inverse and is therefore an h-equivalence. From fg'id we then conclude that f is an h-equivalence. The importance of the last theorem lies in the fact that “homotopy equivalence” can be tested algebraically. Note that the theorem does not say: If q.Y / Š q.Z/ for each q, then Y and Z are homotopy equivalent; it is important to have a map which induces an isomorphism of homotopy groups. Mapping a space to a point gives: (8.4.4) Corollary. A CW-complex X is contractible if and only if q.X/ D 0 for q 0. (8.4.5) Example. From j.S n/ D 0 for j < n and j.S 1/ D colimn j.S n/ we conclude that the homotopy groups of S 1 are trivial. Hence S 1 is contractible.Þ (8.4.6) Example. A simply connected 1-dimensional complex is contractible. A Þ contractible 1-dimensional CW-complex is called a tree. (8.4.7) Theorem. A connected CW-complex X contains a maximal (with respect to inclusion) tree as subcomplex. A tree in
X is maximal if and only if it contains each 0-cell. Proof. Let B denote the set of all trees in X, partially ordered by inclusion. Let T B be a totally ordered subset. Then C D T 2T T is contractible: 1.C / D 0, since a compact subset of C is contained in a ﬁnite subcomplex and therefore in some T 2 T. Thus, by Zorn’s lemma, there exist maximal trees. S Let B be a maximal tree. Consider the 1-cells which have at least one end point in B. If the second end point is not contained in B, then B is obviously not maximal. Therefore the union V of these 1-cells together with B form a subcomplex of X 1, and the remaining 1-cells together with their end points form a subcomplex X 1 XV. Since X is connected so is X 1, hence V D X 1, and B 0 D V 0 D X 0. Let B be a tree which contains X 0. Let B 0 B be a strictly larger tree. Since B is contractible, B 0 and B 0=B are h-equivalent. Hence B 0=B is contractible. S 1 and is not simply connected. Since X 0 B, the space B 0=B has the form Contradiction. W 210 Chapter 8. Cell Complexes We now generalize the suspension theorem (6.10.4). Let X and Y be pointed spaces. We have the suspension map † W ŒX; Y 0! Œ†X; †Y 0. We use the adjunction Œ†X; †Y 0 Š ŒX; †Y 0. The resulting map ŒX; Y 0! ŒX; †Y 0 is then induced by the pointed map W Y! †Y which assigns to y 2 Y the loop t 7! Œy; t in †Y. (8.4.8) Theorem. Suppose i.Y / D 0 for 0 i n. Then the suspension † W ŒX; Y 0! Œ†X; †Y 0 is bijective (surjective) if X is a CW-complex of dimension dim X 2n (dim X 2n C 1). Proof. By the suspension theorem (6.10.4), the map is.2n C 1/-connected.
Now use the pointed version of (8.4.2). (8.4.9) Theorem. Let X be a ﬁnite pointed CW-complex. Then † W Œ†kX; †kY 0! Œ†kC1X; †kC1Y 0 is bijective for dim.X/ k 1. Proof. We have dim †kX D k C dim X. The space †Y is path connected. By the theorem of Seifert and van Kampen, †2Y is simply connected. From the suspension theorem we conclude that j.†kY / D 0 for 0 j k 1. By the previous theorem, † is a bijection for k C dim X 2.k 1/. 8.5 Cellular Approximation (8.5.1) Proposition. Suppose X is obtained from A by attaching.n C 1/-cells. Then.X; A/ is n-connected. Proof. We know that.DnC1; S n/ is n-connected. Now apply (6.4.2). (8.5.2) Proposition. Let X be obtained from A by attaching n-cells (n 1). Suppose A is simply connected. Then the quotient map induces an isomorphism n.X; A/! n.X=A/. Proof. (8.5.1) and (6.10.2). (8.5.3) Proposition. For each relative CW-complex.X; A/ the pair.X; X n/ is n-connected. Proof. From (8.5.1) we obtain by induction on k that.X nCk; X n/ is n-connected. The compactness argument (8.3.8) ﬁnally shows.X; X n/ to be n-connected. Let X and Y be CW-complexes. A map f W X! Y is cellular, if f.X n/ Y n for each n 2 N0. The cellular approximation theorem (8.5.4) is an application of (8.4.1). 8.6. CW-Approximation 211 (8.5.4) Theorem. A map f W X! Y is homotopic to a cellular map g W X! Y. If B X is a subcomplex and f jB cellular, then
the homotopy f'g can be chosen relative to B. Proof. We show inductively that there exist homotopies H n W X I! Y such that 1 D H n 0 D f, H n1 (1) H 0 (2) H n 1.X i / Y i for i n; (3) H n is constant on X n1 [ B. 0 for n 1; For the induction step we assume f.X i / Y i for i < n. Let ˆ W.Dn; S n1/!.X n; X n1/ be a characteristic map of an n-cell not contained in B. The map f ı ˆ is homotopic relative to S n1 to a map into Y n, since.Y; Y n/ is n-connected. A corresponding homotopy is used to deﬁne a homotopy of f on the associated closed n-cells. This process deﬁnes the homotopy on B [ X n; and we extend it to X, using the fact that B [ X n X is a subcomplex and hence a coﬁbration. We now concatenate the homotopies H n: ( H.x; t/ D H i.x; 2iC1.t 1 C 2i //; H i.x; 1/; 1 2i t 1 2i1; x 2 X i ; t D 1: This map is continuous on X i I and hence on X I, since this space is the colimit of the X i I. (8.5.5) Corollary. Let f0; f1 W X! Y be cellular maps which are homotopic. Then there exists a homotopy f between them such that f.X n I / Y nC1. If f0; f1 are homotopic rel B, then f can be chosen rel B. Proof. Choose a homotopy f W f0'f1 rel B. Then f maps X @I [ B I into Y n. Now apply (8.5.4) toX @I [ B I X I. Problems 1. Let A X be a subcomplex and f W A! Y a cellular map. Then Y D X [f Y is a CW-complex. 2. A CW-complex is path connected if and only
if the 1-skeleton is path connected. The components are equal to the path components, and the path components are open. 8.6 CW-Approximation We show in this section, among other things, that each space is weakly homotopy equivalent to a CW-complex. Our ﬁrst aim is to raise the connectivity of a map. (8.6.1) Theorem. Let f W A! Y be a k-connected map, k 1. Then there exists for each n > k a relative CW-complex.X; A/ with cells only in dimensions 212 Chapter 8. Cell Complexes j 2 fk C 1; : : : ; ng, n 1, and an n-connected extension F W X! Y of f. If A is CW-complex, then A can be chosen as a subcomplex of X. Proof. (Induction over n.) Recall that the map f is k-connected if the induced map f W j.A; /! j.Y; f.// is bijective for j < k and surjective for j D k (no condition for k D 1). If we attach cells of dimension greater than k and extend, then the extension remains k-connected. This fact allows for an inductive construction. Let n D 0, k D 1. Suppose f W 0.A/! 0.Y / is not surjective. Let C D fcj j j 2 J g be a family of points in Y which contains one element from each path component 0.Y / n f0.A/. Set X D A C j and deﬁne F W X! Y by F jA D f and F.D0 j / D fcj g. Then X is obtained from A by attaching 0-cells and F is a 0-connected extension of f. D0 ` n D 1. Suppose f W A! Y is 0-connected. Then f W 0.A/! 0.Y / is surjective. Let c1; c1 be points in different path components of A which have the same image under f. Then'W S 0! A, '.˙1/ D c˙1, is an attaching map for a 1-cell. We can extend f over A [' D1 by a path from f.c/ to f.cC/. Treating other pairs of path components similarly, we obtain
an extension F 0 W X 0! Y of f over a relative 1-complex.X 0; A/ such that F 0 W 0.X 0/! 0.Y / is bijective. The bijectivity of F 0 j D f with the inclusion j W A! X 0; the map j is 0-connected; path components with the same image under f have, by construction, the same image under j. follows from these facts: We have F 0 We still have to extend F 0 W X 0! Y to a relative 1-complex X X 0 such that F W 1.X; x/! 1.Y; f.x// is surjective for each x 2 X. Let Fj W.D1; S 0/!.Y; y/ be a family of maps such that the ŒFj 2 1.Y; y/ together with F 0.1.X 0; x// generate 1.Y; y/, y D F 0.x/. Let X X 0 be obtained from X 0 by attaching 1cells with characteristic maps.ˆj ; 'j / W.D1; S 0/!.X 0; x/. We extend F 0 to F such that F ı ˆj D Fj. Then F W 1.X; x/! 1.Y; y/ is surjective. n 2. Suppose f W A! Y is.n 1/-connected. By the use of mapping cylinders, we can assume that f is an inclusion. Let.ˆj ; 'j / W.Dn; S n1; e0/!.Y; A; a/ be a set of maps such that the yj D Œˆj ; 'j 2 n.Y; A; a/ generated the 1.A; a/-module n.Y; A; a/. We attach n-cells to A by attaching maps 'j to obtain X and extend f to F by the null homotopies ˆj of f 'j. The characteristic map of the n-cell with attaching map 'j represents xj 2 n.X; A; a/ and Fxj D yj. The map F induces a morphism of the exact homotopy sequence of.X; A; a/ into the sequence of.Y; A; a/, and F W n.X; A; a/! n
.Y; A; a/ is surjective by construction. Consider the diagram n.A/ D n.A/ n.X/.1/ F n.Y / n.X; A/ n1.A/ n1.X/.2/ F n.Y; A/ D n1.A/.3/ F n1.Y / 0 0. 8.6. CW-Approximation 213 The sequences end with 0, since n1.X; A/ D 0 and n1.A/! n1.Y / is surjective by assumption. (2) is surjective. The Five Lemma shows us that (1) is surjective and (3) injective. By induction hypothesis, (3) is already surjective. Hence F is n-connected. In order to obtain A as a subcomplex of X, one works with cellular attaching maps. (8.6.2) Theorem. Let Y be a CW-complex such that i.Y / D 0 for 0 i k. Then Y is homotopy equivalent to a CW-complex X with X k D fg. Proof. Start with the k-connected map f W A D fg! Y and extend it to a weak equivalence F W X! Y by attaching cells of dimension greater than k. (8.6.3) Proposition. Let A and B be pointed CW-complexes. Assume that A is.m 1/-connected and B is.n 1/-connected. Then A ^k B is.m C n 1/connected. Proof. We can assume that A has no cells in dimensions less than m and n no cells in dimensions less than n (except the base point). Then A ^k B has no cells in dimensions less than m C n. (8.6.4) Theorem. Let.Xj j j 2 J / be a family of.n1/-connected CW-complexes. Let k W Xk! j 2J Xj be the inclusion of the k-th summand. Then W L j 2J n.Xj /! n j 2J Xj W ˛J D hj W i is an isomorphism (n 2). Proof. Let J be ﬁnite. Up to h-equivalence we can assume that Xj has no cells in dimensions less than n, except the base
point. Then j Xj is obtained from W j Xj by attaching cells of dimension 2n. Hence m Xj is Q W! m Q Xj an isomorphism for m 2n 2. From the diagram Q W n Xj Š n Xj ˛J L n.Xj /.1/ Š Q Š.pj / n.Xj / we conclude that ˛J is an isomorphism. W there exists a ﬁnite E J Let now J be arbitrary. For each x 2 n such that x is contained in the image of n J Xj! n, since a compact subset is contained in a ﬁnite wedge. The result for E now shows that ˛J is surjective. If x1 and x2 have the same image under ˛J, then these elements E and, again by a compactness argument, they are contained in some ﬁnite sum have the same image under some ˛E, ifE is chosen large enough. This shows the injectivity of ˛J. j 2J Xj W E Xj W L 214 Chapter 8. Cell Complexes (8.6.5) Proposition. Suppose j.Y / D 0 for j > n. Let X be obtained from A by attaching cells of dimension n C 2. Then A X induces a bijection ŒX; Y! ŒA; Y. Proof. Surjective. Let f W A! Y be given. Attach.n C 2/-cells via maps'W S nC1! A. Since f'W S nC1! Y is null homotopic, we can extend f over the.n C 2/-cells. Continue in this manner. Injective. Use the same argument for.X I; X @I [ A I /. The cells of this relative complex have a dimension > n C 2. (8.6.6) Theorem. Let A be an arbitrary space and k 2 N0. There exists a relative CW-complex.X; A/ with cells only in dimensions j kC2, such that n.X; x/ D 0 for n > k and x 2 X, and the induced map n.A; a/! n.X; a/ is an isomorphism for n k and a 2 A. Proof. We construct inductively for t
2 a sequence A D X kC1 X kC2 X kCt such that n.A; a/ Š n.X kCt ; a/ for n k, n.X kCt ; a/ D 0 for k < n k C t 1, and X mC1 is obtained from X m by attaching.m C 1/-cells. The induction step: If we attach.m C 1/-cells to X m by the attaching maps j ; e0/!.X m; a/ to obtain X mC1, then n.X m; a/ Š n.X mC1; a/ for 'j W.S m n m 1. The exact sequence mC1.X mC1; X m; a/ @! m.X m; a/! m.X mC1; a/! 0 shows that the Œ'j are in the image of @. Thus, if the Œ'j generate m.X m; a/, then m.X mC1; a/ D 0. (8.6.7) Example. We can attach cells of dimension n C 2 to S n to obtain a space K.Z; n/ which has a single non-trivial homotopy group n.K.Z; n// Š Z. See Þ the section on Eilenberg–Mac Lane spaces for a generalization. Let i X n induces an isomorphism k.i X n W X! XŒn be an inclusion of the type constructed in (8.6.6), namely XŒn is obtained by attaching cells of dimension greater than n C 1 such that k.XŒn/ D 0 for k > n and i X n / for k n. Given a map f W X! Y and i Y m W Y! Y Œm for m n, there exists a unique homotopy class fn;m W XŒn! Y Œm such that i Y n ; this is a consequence of (8.6.5). We let j X n. We call j X n the n-connective covering of X. The induced map i.j X n / W i.Xhni/! i.X/ is an isomorphism for i > n and i.Xhni/ D 0 for i n. The universal covering has such properties in the
case that n D 1. So we have a generalization, in the realm of ﬁbrations. Objects of this type occur in the theory of Postnikov decompositions of a space, see e.g., [192]. n W Xhni!X be the homotopy ﬁbre of i X m ı f D fn;m ı i X As a consequence of (8.6.1) for A D ; we see that for each space Y there exists a C W -complex X and a weak equivalence f W X! Y. We call such a weak 8.6. CW-Approximation 215 equivalence a CW-approximation of Y. Note that a weak equivalence between CWcomplexes is a homotopy equivalence (8.4.3). We show that CW-approximations are unique up to homotopy and functorial in the homotopy category. (8.6.8) Theorem. Let f W Y1! Y2 be a continuous map and let ˛j W Xj! Yj be CW-approximations. Then there exists a map'W X1! X2 such that f ˛1'˛2', and the homotopy class of'is uniquely determined by this property. Proof. Since ˛2 is a weak equivalence, ˛2 W ŒX1; X2! ŒX1; Y2 is bijective. Hence there exists a unique homotopy class'such that f ˛1'˛2'. A domination of X by K consists of maps i W X! K; p W K! X and a homotopy pi'id.X/. (8.6.9) Proposition. Suppose M is dominated by a CW-complex X. Then M has the homotopy type of a CW-complex. Proof. Suppose i W M! X and r W X! M are given such that ri is homotopic to the identity. There exists a CW-complex W X Y and an extension R W Y! M of r such that R induces an isomorphism of homotopy groups. Let j D i W M! X! Y. Since Rj D ri'id, the composition Rj induces isomorphisms of homotopy groups, hence so does j. From jRj '
j we conclude that jR induces the identity on homotopy groups and is therefore a homotopy equivalence. Let k be h-inverse to jR, then j.Rk/'id. Hence j has the left inverse R and the right inverse Rk and is therefore a homotopy equivalence. A (half-exact) homotopy functor on the category C 0 of pointed connected CWspaces is a contravariant functor h W C 0! SET0 into the category of pointed sets with the properties: (1) (Homotopy invariance) Pointed homotopic maps induce the same morphism. (2) (Mayer–Vietoris property) Suppose X is the union of subcomplexes A and B. If a 2 h.A/ and b 2 h.B/ are elements with the same restriction in h.A\B/, then there exists an element x 2 h.X/ with restrictions a and b. j Xj with inclusions ij W Xj! X. Then (3) (Additivity) Let X D W h.X/! Q j h.Xj /; x 7!.h.ij /x/ is bijective. (8.6.10) Theorem (E. H. Brown). For each homotopy functor h W C 0! SET0 there exist K 2 C 0 and u 2 h.K/ such that ŒX; K0! h.X/; Œf 7! f.u/ is bijective for each X 2 C 0. 216 Chapter 8. Cell Complexes In category theory one says that K is a representing object for the functor h. The theorem is called the representability theorem of E. H. Brown. For a proof see [31], [4]. (8.6.11) Example. Let h.X/ D ŒX; Z0 for a connected pointed space Z. Then h is a homotopy functor. From (8.6.10) we obtain K 2 C 0 and f W K! Z such that f W ŒX; K0! ŒX; Z0 is always bijective, i.e., f is a weak h-equivalence. Thus Þ we have obtained a CW-approximation X of Z. Problems 1. As a consequence of (8.6.8)
one can extend homotopy functors from CW-complexes to arbitrary spaces. Let F be a functor from the category of CW-complexes such that homotopic maps f'g induce the same morphism F.f / D F.g/. Then there is, up to natural isomorphism, a unique extension of F to a homotopy invariant functor on TOP which maps weak equivalences to isomorphisms. 2. A point is a C W -approximation of the pseudo-circle. 3. Determine the C W -approximation of f0g [ fn1 j n 2 Ng. 4. Let X and Y be CW-complexes. Show that the identity X k Y! X Y is a CWapproximation. 5. Let.Yj j j 2 J / be a family of well-pointed spaces and ˛j W Xj! Yj a family of j ˛j is a CW-approximation. Give a counterexample pointed CW-approximations. Then (with two spaces) in the case that the spaces are not well-pointed. 6. Let f W A! B and g W C! D be pointed weak homotopy equivalences between wellpointed spaces. Then f ^ g is a weak homotopy equivalence. 7. Verify from the axioms of a homotopy functor that h.P / for a point P contains a single element. 8. Verify from the axioms of a homotopy functor that for each inclusion A X in C 0 the canonical sequence h.X=A/! h.X/! h.A/ is an exact sequence of pointed sets. W 8.7 Homotopy Classiﬁcation In favorable cases the homotopy class of a map is determined by its effect on homotopy groups. (8.7.1) Theorem. Let X be an.n 1/-connected pointed CW-complex. Let Y be a pointed space such that i.Y / D 0 for i > n 2. Then hX W ŒX; Y 0! Hom.n.X/; n.Y //; Œf 7! f is bijective. Proof. The assertion only depends on the pointed homotopy type of X. We use (8.6.2) and assume X n1 D fg. The h
X constitute a natural transformation in the variable X. Since.X; X nC1/ is.nC1/-connected, the inclusion X nC1 X induces 8.8. Eilenberg–Mac Lane Spaces 217 an isomorphism on n. By (8.6.5), the restriction r W ŒX; Y 0! ŒX nC1; Y 0 is a bijection. Therefore it sufﬁces to consider the case, that X has, apart from the base point, only cells of dimension n and n C 1. Moreover, by the homotopy theorem for coﬁbrations, we can assume that the attaching maps for the.nC1/-cells are pointed. S n In this case X is the mapping cone of a pointed map f W A D j D B. We therefore have the exact coﬁbre sequence ŒA; Y 0 f Our assumption about Y yields Œ†A; Y 0 D We apply the natural transformation h and obtain a commutative diagram ŒB; Y 0 ŒX; Y 0 Œ†A; Y 0: k nC1.Y / D 0. S n k! 0 Š †S n k ; Y W Q W W ŒA; Y 0 hA f ŒB; Y 0 hB f 1 ŒX; Y 0 hX Hom.nA; nY / Hom.nB; nY / Hom.nX; nY / 0 0. As one of the consequences of the excision theorem we showed that the sequence n.A/! n.B/! n.X/! 0 is exact, and therefore the bottom sequence of the diagram is exact. We show that hA and hB are isomorphisms. If A D S n, then hA W n.Y / D ŒS n; Y 0! Hom.n.S n/; n.Y // W is an isomorphism, since n.S n/ is generated by the identity. A D S n k, we have a commutative diagram In the case that W 0 Š Q S n k ; Y hW Sn k ŒS n k ; Y Q Š hSn k W n Hom ; n.Y / S n k.1/ Q Hom.n.S n L k /; n.Y //. W n.
cells of dimensions n C 2 to X in order to obtain a K.; n/, see (8.6.6). (8.8.2) Examples. The space S 1 is a K.Z; 1/. We know 1.S 1/ Š Z, and from the exact sequence of the universal covering R! S 1 we know that n.S 1/ D 0 for n 2. The space CP 1 is a model for K.Z; 2/. The space RP 1 is a K.Z=2; 1/. Þ The adjunction Œ†X; Y 0 Š ŒX; Y 0 shows that K.; n C 1/ has the homotopy groups of a K.; n/. By a theorem of Milnor [132], [67], Y has the homotopy type of a CW-complex if Y is a CW-complex. If one does not want to use this result one has the weaker result that there exists a weak homotopy equivalence K.; n/! K.; n C 1/. We now establish further properties of Eilenberg–Mac Lane spaces. We begin by showing that Eilenberg–Mac Lane spaces are H -spaces. Then we construct product pairings K.; m/ ^ K.; n/! K. ˝ ; m C n/. In this context ˝ denotes the tensor product of the abelian groups and (alias Z-modules) over Z. We call the space K.; n/ polarized, if we have chosen a ﬁxed isomorphism ˛ W n.K.; n//!. If.K.; n/; ˛/ and.K.; n/; ˇ/ are polarized complexes, the product K.; n/ K.; n/ will be polarized by n.K.; n/ K.; n// Š n.K.; n// n.K.; n// ˛ˇ 1 2: (8.8.3) Proposition. Having chosen polarizations, we obtain from (8.7.1) an iso- morphism ŒK.; n/; K.; n/0 Š Hom.; /. 8.8. Eilenberg–Mac Lane Spaces 219 (8.8.4) Theorem. Let be an abelian group. Then an Eilenberg–Mac Lane complex K.; n/ is a commutative group object in h-
TOP. Proof. Let K D.K.; n/; ˛/ be a polarized complex with base point a 0-cell e. For an abelian group, the multiplication W!,.g; h/ 7! gh is a homomorphism. Therefore there exists a map m W K K! K, unique up to homotopy, which corresponds under (8.8.3) to. Similarly, W!, g 7! g1 is a homomorphism and yields a map i W K! K. Claim:.K; m; i/ is an associative and commutative H -space. The maps m ı.m id/ and m ı.id m/ induce the same homomorphism when n is applied; hence these maps are homotopic. In a similar manner one shows that x 7! m.x; e/ is homotopic to the identity. Since K _ K K K is a coﬁbration, we can change m by a homotopy such that m.x; e/ D m.e; x/ D x. We write x 7! m.x; i.x// as composition d! K K id i! K K m! K K and apply n; the result is the constant homomorphism. Hence this map is null homotopic. Commutativity is veriﬁed in a similar manner by applying n. See also Problem 1. For the construction of the product pairing we need a general result about products for homotopy groups. We take the smash product of representatives f W I m=@I m! X, g W I n=@I n! Y and obtain a well-deﬁned map m.X/ n.Y /! mCn.X ^k Y /;.Œf ; Œg/ 7! Œf ^ g D Œf ^ Œg: We call this map the ^-product for homotopy groups. It is natural in the variables X and Y. (8.8.5) Proposition. The ^-product is bi-additive. Proof. The additivity in the ﬁrst variable follows directly from the deﬁnition of the addition, if we use C1. We see the additivity in the second variable, if we use the composition laws C1 and CmC
1 in the homotopy groups. (8.8.6) Proposition. Let A be an.m 1/-connected and B an.n 1/-connected CW-complex. Then A ^ B is.m C n 1/-connected and the ^-product m.A/ ˝ n.B/! mCn.A ^k B/ is an isomorphism (m; n 2). If m or n equals 1, then one has to use the abelianized groups. 220 Chapter 8. Cell Complexes Proof. The assertion about the connectivity was shown in (8.6.3). For A D S m the assertion holds by the suspension theorem. For A D j we use the commutative diagram.B/ ^ mCn W S m j ^k B L.1/ m.S m j / ˝ n.B/.2/ W.S m j ^ B/ mCn.3/.4/ L.m.S m j / ˝ n.B// L.5/ mCn.S m j ^ B/. (1) is an isomorphism by (8.6.4). (2) is induced by a homeomorphism. (3) is an isomorphism by algebra. (4) is an isomorphism by (8.6.4). (5) is an isomorphism by the suspension theorem. This settles the case of a wedge of m-spheres. Next we let A be the mapping cone of a map f W C! D where C and D are wedges of m-spheres. Then we have a commutative diagram with exact rows: m.C / ˝ m.Y / m.D/ ˝ n.B/ m.A/ ˝ n.B/ Š Š mCn.C ^k B/ mCn.D ^k B/ mCn.A ^k B/ 0 0. The general case now follows from the observation that the inclusion AmC1! A induces an isomorphism on m and AmC1 ^k B! A^k B induces an isomorphism on mCn. Let.K.G; m/; ˛/,.K.H; n/; ˇ/ and.K.G ˝ H; m C n/; / be polarized E
ilen- berg–Mac Lane complexes for abelian groups G and H. A product is a map m;n W K.G; m/ ^k K.H; n/! K.G ˝ H; m C n/ such that the diagram m.K.G; m// ˝ n.K.H; n// ^ mCn.K.G; m/ ^k K.H; n// ˛˝ˇ G ˝ H.m;n/ mCn.K.G ˝ H; m C n// is commutative. Here we have use the ^-product (8.8.5). (8.8.7) Theorem. There exists a product. It is unique up to homotopy. 8.8. Eilenberg–Mac Lane Spaces 221 Proof. Let G and H be abelian groups. The ﬁrst non-trivial homotopy group of K.G; m/ ^k K.H; n/ is mCn and it is isomorphic to G ˝ H, see (8.8.6). By (8.6.6) there exists an inclusion m;n W K.G; m/ ^k K.H; n/! K.G ˝ H; m C n/: We can choose the polarization so that the diagram above becomes commutative. Uniqueness follows from (8.7.1). The products (8.8.7) are associative, i.e., mCn;p ı.m;n id/'m;nCp ı.id n;p/: The products are graded commutative in the following sense: K.; m C n/ ı m;n '.1/mn 0 ı n;m with the interchange maps 0 W K.m; G/ ^ K.n; H /! K.n; H / ^ K.m; G/ and W G ˝ H! H ˝ G. Let R be a commutative ring with 1. We think of the multiplication as being a homomorphism W R˝R! R between abelian groups. From this homomorphism we obtain a unique homotopy class K./ W K.R ˝ R; m/! K.
R; m/. We compose K./ with k;l and obtain a product map mk;l W K.R; k/ ^k K.R; l/! K.R; k C l/: Also these products are associative and graded commutative. In the associated homotopy groups H k.XI R/ D ŒX C; K.R; k/0 we obtain via.f; g/ 7! mk;l.f ^ g/ products H k.XI R/ ˝ H l.Y I R/! H kCl.X Y I R/; which are also associative and graded commutative. (See also Problem 3.) In a similar manner we can start from an R-module structure R ˝ M! M on M. Later, when we study singular cohomology, we show that for a CW-complex X the group ŒX C; K.R; k/0 is naturally isomorphic to the singular cohomology group H k.XI R/ with coefﬁcients in the ring R. This opens the way to a homotopical study of cohomology. The product (8.8.7) can then be used to construct the so-called cup product in cohomology. Once singular cohomology theory is constructed one obtains from the representability theorem of Brown Eilenberg–Mac Lane spaces as representing objects. 8.8.8 Eilenberg–Mac Lane spectra. Let A be an abelian group. The Eilenberg– Mac Lane spectrum HA consists of the family.K.A; n/ j n 2 N0/ of Eilenberg– Mac Lane CW-spaces and maps en W †K.A; n/! K.A; n C 1/ (an inclusion of 222 Chapter 8. Cell Complexes subcomplexes; attach cells to †K.A; n/ to obtain a K.A; n C 1/). This spectrum is an -spectrum. We have proved in any case that "n W K.A; n/! K.A; n C 1/ is a weak homotopy equivalence. This sufﬁces if one wants to deﬁne the cohomology Þ theory only for pointed CW-spaces. Problems 1. From the natural isomorphism ŒX; K
.; n/0 Š ŒX; 2K.; n C 2/0 we see that X 7! ŒX; K.; n/0 is a contravariant functor into the category of abelian groups. Therefore, by category theory, there exists a unique (up to homotopy) structure of a commutative h-group on K.; n/ inducing the group structures of this functor. 2. Let ˛ 2 m.X/, ˇ 2 n.Y /, and W X ^k Y! Y ^k X the interchange map. Then ˛ ^ ˇ D.1/mnˇ ^ ˛. 3. Let M be an R-module. A left translation lr W M! M, x 7! rx is a homomorphism of the abelian group M and induces therefore a map Lr W K.M; k/! K.M; k/. Use these maps to deﬁne a natural structure of an R-module on ŒX; K.M; k/0. 4. The simply connected surfaces are S 2 and R2 [44, p. 87]. If a surface is different from S 2 and RP 2, then it is a K.; 1/. 5. Let E./! B./ be a -principal covering with contractible E./. Then B./ is a K.; 1/. Spaces of the type B./ will occur later as classifying spaces. There is a bijection ŒK.; 1/; K.; 1/ Š Hom.; /= between homotopy classes and group homomorphisms up to inner automorphisms. 6. Let S 1 be the colimit of the unit spheres S.Cn/ S.CnC1/. This space carries a free action of the cyclic group Z=m S 1 by scalar multiplication. Show that S 1 with this action is a Z=m-principal covering. The quotient space is a CW-space B.Z=m/ and hence a K.Z=m; 1/. 7. A connected 1-dimensional CW-complex X is a K.; 1/. Determine from the topology of X. 8. A connected non-closed surface (with or without boundary) is a K.; 1/. Chapter 9 Singular Homology Homology is the most ingenious invention in algebraic topology
. Classically, the deﬁnition of homology groups was based on the combinatorial data of simplicial complexes. This deﬁnition did not yield directly a topological invariant. The deﬁnition of homology groups and (dually) cohomology groups has gone through various stages and generalizations. The construction of the so-called singular homology groups by Eilenberg [56] was one of the deﬁnitive settings. This theory is very elegant and almost entirely algebraic. Very little topology is used as an input. And yet the homology groups are deﬁned for arbitrary spaces in an invariant manner. But one has to pay a price: The deﬁnition is in no way intuitively plausible. If one does not mind jumping into cold water, then one may well start algebraic topology with singular homology. Also interesting geometric applications are easily obtainable. In learning about homology, one has to follow three lines of thinking at the same time: (1) The construction. (2) Homological algebra. (3) Axiomatic treatment. (1) The construction of singular homology groups and the veriﬁcation of its main properties, now called the axioms of Eilenberg and Steenrod. (2) A certain amount of algebra, designed for use in homology theory (but also of independent algebraic interest). It deals with diagrams, exact sequences, and chain complexes. Later more advanced topics are needed: Tensor products, linear algebra of chain complexes, derived functors and all that. (3) The object that one constructs with singular homology is now called a homology theory, deﬁned by the axioms of Eilenberg and Steenrod. Almost all applications of homology are derived from these axioms. The axiomatic treatment has other advantages. Various other homology and cohomology theories are known, either constructed by special input (bordism theories, K-theories, de Rham cohomology) or in a systematic manner via stable homotopy and spectra. The axioms of a homology or cohomology theory are easily motivated from the view-point of homotopy theory. But we should point out that many results of algebraic topology need the idea of homology: The reduction to combinatorial data via cell complexes, chain
complexes, spectral sequences, homological algebra, etc. Reading this chapter requires a parallel reading of the chapter on homological algebra. Already in the ﬁrst section we use the terminology of chain complexes and their homology groups and results about exact sequences of homology groups. 224 Chapter 9. Singular Homology 9.1 Singular Homology Groups The n-dimensional standard simplex is n D Œn D ˚.t0; : : : ; tn/ 2 RnC1 P ˇ ˇ n iD0 ti D 1; ti 0 RnC1:.˛/ W Œm! Œn; We set Œn D f0; : : : ; ng. A weakly increasing map ˛ W Œm! Œn induces an afﬁne map P m iD0 ti e˛.i/: m iD0 ti ei D.t0; : : : ; tm/. These maps Here ei is the standard unit vector, thus satisfy the rules of a functor.˛ ı ˇ/ D.˛/ ı.ˇ/ and.id/ D id. Let ın i W Œn 1! Œn be the injective map which misses the value i. m iD0 ti ei 7! P P (9.1.1) Note. ınC1 write d n i D.ın rules. j i i D ınC1 ın ın j 1, i < j. (The composition misses i and j.) We i /. By functoriality, the d n i satisfy the analogous commutation A continuous map W n! X is called a singular n-simplex in X. The i -th face of is ı d n i. We denote by Sn.X/ the free abelian group with basis the set of singular n-simplices in X. (We also set, for formal reasons, Sn.X/ D 0 in the case that n < 0 but disregard mostly this trivial case. If X D ;, we let Sn.X/ D 0.) An element x 2 Sn.X/ is called a singular n-chain. We think of x n, n 2 Z. In practice, we skip as a formal ﬁnite linear combination x D
of S.X/ induces a boundary operator @n W Sn.X; A/! Sn1.X; A/ such that the family of quotient homomorphisms Sn.X/! Sn.X; A/ is a chain map. The homology groups Hn.X; A/ D Hn.X; AI Z/ of S.X; A/ are the relative singular homology groups of the pair.X; A/ (with coefﬁcients in Z). A continuous map f W.X; A/!.Y; B/ induces a chain map f W S.X; A/! S.Y; B/ and homomorphisms f D Hq.f / W Hq.X; A/! Hq.Y; B/. In this way, Hq becomes a functor from TOP.2/ to ABEL. We apply (11.3.2) to the exact sequence of singular chain complexes 0! S.A/! S.X/! S.X; A/! 0 and obtain the associated exact homology sequence: (9.1.3) Theorem. For each pair.X; A/ the sequence @! Hn.A/! Hn.X/! Hn.X; A/ @! Hn1.A/! is exact. The sequence terminates with H0.X/! H0.X; A/! 0. The undecorated arrows are induced by the inclusions.A; ;/.X; ;/ and.X; ;/.X; A/. Let.X; A; B/ be a triple, i.e., B A X. The inclusion S.A/! S.X/ induces by passage to factor groups an inclusion S.A; B/! S.X; B/, and its cokernel can be identiﬁed with S.X; A/. We apply (11.3.2) to the exact sequence of chain complexes 0! S.A; B/! S.X; B/! S.X; A/! 0 and obtain the exact sequence of a triple @! Hn.A; B/! Hn.X; B/! Hn.X; A/ @! Hn1.A; B/! : The boundary operator @ W Hn.X; B/! Hn
1.A; B/ in the exact sequence of a triple is the composition of the boundary operator for.X; A/ followed by the map Hn1.A/! Hn1.A; B/ induced by the inclusion. It remains to verify that the connecting morphisms @ constitute a natural transformation, i.e., that for each map between triples f W.X; A; B/!.X 0; A0; B 0/ the 226 Chapter 9. Singular Homology diagram Hk.X; A/ f Hk.X 0; A0/ @ @ Hk1.A; B/ f Hk1.A0; B 0/ is commutative. This is a special case of an analogous fact for morphisms between short exact sequences of chain complexes and their associated connecting morphisms. We leave this as an exercise. One cannot determine the groups Hq.X; A/ just from its deﬁnition (except in a few trivial cases). Note that for open sets in Euclidean spaces the chain groups have an uncountable basis. So it is clear that the setup only serves theoretical purposes. Before we prove the basic properties of the homology functors (the axioms of Eilenberg and Steenrod) we collect a few results which follow directly from the deﬁnitions. 9.1.4 Point. Let X D P be a point. There is a unique singular n-simplex, hence Sn.P / Š Z, n 0. The boundary operators @0 and @2iC1 are zero and @2; @4; : : : 6D 0; and H0.P / Š Z, via the are isomorphisms. Hence Hi.P / D 0 for i Þ homomorphism which sends the unique 0-simplex to 1 2 Z. 9.1.5 Additivity. Let.Xj j W.Xj ; Xj \ A/!.X; A/ be the inclusion. Then j j 2 J / be the path components of X, and let L j 2J Sn.Xj ; Xj \ A/! Sn.X; A/;.xj / 7! P j j #.xj / is an isomorphism. Similarly for Hn instead of Sn. The reason is that n is path connected, and therefore W n! X has an image in
one of the Xj, so we can Þ sort the basis elements of Sn.X/ according to the components Xj. 9.1.6 The groups H0. The group H0.X/ is canonically isomorphic to the free abelian group Z0.X/ over the set 0.X/ of path components. We identify a singular 0-simplex W 0! X with the point.0/. Then S0.X/ is the free abelian group on the points of X. A singular 1-simplex W 1! X is essentially the same thing as a path, only the domain of deﬁnition has been changed from I to 1. We associate to the path w W I! X, t 7!.1t; t /. Then @0 D w.1/ and @1 D w.0/, hence @ D @0 @1 corresponds to the orientation convention @w D w.1/ w.0/. If two points a; b 2 X are in the same path component, then the zero-simplices a and b are homologous. Hence we obtain a homomorphism from Z0.X/ into H0.X/, if we assign to the path component of a its homology class. We also have a homomorphism S0.X/! Z0.X/ which sends the singular simplex of a 2 X to the path component of a. This homomorphism sends the image of @ W S1.X/! S0.X/ to zero. Hence we obtain an inverse homomorphism Þ H0.X/! Z0.X/. 9.2. The Fundamental Group 227 9.2 The Fundamental Group The signs which appear in the deﬁnition of the boundary operator have an interpretation in low dimensions. They are a consequence of orientation conventions. A singular 1-simplex W 1! X is essentially the same thing as a path, only the domain of deﬁnition has changed from Œ0; 1 to 1. We associate to the path I! X, t 7!.1 t; t/. The inverse path is then.t0; t1/ D.t1; t0/. The product of paths has now the form (. /.t0; t1/ D.2t0 1; 2t1/;.2t0; 2t1 1
/; t1 1=2; t1 1=2: If we deﬁne! W 2! X,.t0; t1; t2/ 7!. /.t0 C t1=2; t1=2 C t2/, then one veriﬁes @! D C. A loop W 1! X is a 1-cycle; let Œ be its homology class. Thus for loops ; we have.1/ Œ D Œ C Œ: (Here Œz denotes the homology class of the cycle z.) Let k W 1 I! X be a homotopy of paths 1! X. The map k factors over the quotient map q W 1 I! 2,.t0; t1; t/ 7!.t0; t1.1 t/; t1t/ and yields W 2! X. We compute @ D c k1 C k0, with a constant c. A constant 1-simplex is a boundary. Hence k0 k1 is a boundary Œk0 D Œk1 2 C1.X/=B1.X/: In particular, homotopic loops yield the same element in H1.X/. Thus we obtain a well-deﬁned map h0 W 1.X; x0/! H1.X/; by (1), it is a homomorphism. The fundamental group is in general non-abelian. Therefore we modify h0 algebraically to take this fact into account. Each group G has the associated abelianized factor group Gab D G=ŒG; G; the commutator group ŒG; G is the normal subgroup generated by all commutators xyx1y1. A homomorphism G! A to an abelian group A factorizes uniquely over Gab. We apply this deﬁnition to h0 and obtain a homomorphism h W 1.X; x0/ab! H1.X/: (9.2.1) Theorem. Let X be path connected. Then h is an isomorphism. Proof. We construct a homomorphism in the other direction. For x 2 X we choose a path u.x/ from x0 to x. We assign to a 1-simplex W 1! X from 0 D.1; 0/ to 1
D.0; 1/ the class of the loop.u.0/ / u.1/. We extend this assignment linearly to a homomorphism l 0 W C1.X/! 1.X; x0/ab. Let W 2! X be a 2-simplex with faces j D dj. Since 2 is contractible, 2 0'1. This implies l 0.Œ2/ C l 0.Œ0/ D l 0.Œ2 C Œ0/ D l 0.Œ2 0/ D l 0.Œ1/: 228 Chapter 9. Singular Homology Hence l 0 factors over C1.X/=B1.X/ and induces l W H1.X/! 1.X; x0/ab. By a 2 C1.X/ be a construction, lh D id. We show that h is surjective. Let cycle. Then P X X a Œ D a.Œu.0/ C Œ Œu.1// D X a Œ.u.0/ / u.1/; and the last element is contained in the image of h. One of the ﬁrst applications of the homology axioms is the computation H1.S 1/ Š Z. Granted the formal result that 1.S 1/ is abelian, we obtain yet another proof for 1.S 1/ Š Z. 9.3 Homotopy We prove in this section the homotopy invariance of the singular homology groups. We begin with a special case. 9.3.1 Cone construction. Let X be a contractible space. Deﬁne a chain map " D."n/ W S.X/! S.X/ by "n D 0 for n 6D 0 and by "0 0 where 0 W 0! fx0g. We associate to each homotopy h W X I! X from the identity to the constant map with value x0 a chain homotopy s D.sn/ from " to the identity. The homomorphisms s W Sn1.X/! Sn.X/ are obtained from a cone construction. Let n P P n D q W n1 I! n;..0; : : : ; n1/; t/ 7!.t;.1 t/0; : : : ;.
that for continuous X! Y the relations.Nn/.f id/# ı sX n D sY n ı f# hold (naturality). We construct the sn inductively. n D 0. In this case, s0 sends the 0-simplex W 0! fxg X to the 1-simplex s0 W 1! X I,.t0; t1/ 7!.x; t1/. Then the computation @.s0/ D.s0/d0.s0/d1 D 1 0 0 0 shows that.K0/ holds, and also.N0/ is a direct consequence of the deﬁnitions. Now suppose that the sk for k < n are given, and that they satisfy.Kk/ and.Nk/. The identity of n is a singular n-simplex; let n 2 Sn.n/ be the corresponding element. The chain to be constructed snn should satisfy @.snn/ D 1 n.n/ 0 n.n/ sn1@.n/: The right-hand side is a cycle in Sn.n I /, as the next computation shows. @.1 n.n/ 0 D 1 D 1 n.n/ sn1@.n// n1.@n/ 0 n1.@n/ 0 n1.@n/ @sn1.@n/ n1.@n/.1 n1.@n/ 0 n1.@n/ sn2@@n/ D 0: We have used the relation.Kn1/ for @sn1@.n/ and that the t are chain maps. Since n I is contractible, there exists, by (9.3.2), an a 2 SnC1.n I / with the property @a D 1 n.n/ sn1@.n/. We choose an a with this property and deﬁne sn.n/ D a and in general sn./ D. id/#a for W n! X; the required n.n/ 0 230 Chapter 9. Singular Homology naturality.Nn/ forces us to do so. We now verify.Kn/ and.Nn/. We compute @sn./ D @. id/#a D. id/#@a D. id/#.1 n#n 0 D 1 n 0 D 1 n sn1@: nn
sn1@n/ nn 0 n#n sn1#@n We have used:. id/# is a chain map; choice of a; naturality of 1.Nn1/; #n D ; # is a chain map. Thus we have shown.Kn/. The equalities, 0, and.f id/#sn./ D.f id/#. id/#a D.f id/#a D sn.f / D snf# ﬁnally show the naturality.Nn/. With (9.3.3) we control the universal situation. Let f W.X; A/ I!.Y; B/ be a homotopy in TOP.2/ from f 0 to f 1. The sn in (9.3.3) induce by naturality also a chain homotopy sn W Sn.X; A/! SnC1.X I; A I /. The computation # f 0 @.f# ı sn/ C.f# ı sn1/@ D f#@sn C f#sn1@ D f#.1 0/ D f 1 # # to f 1 #. Altogether we see: proves the f#sn to be a chain homotopy from f 0 (9.3.4) Theorem. Homotopic maps induce homotopic chain maps and hence the same homomorphisms between the homology groups. (9.3.5) Example. Let a0.X/; a1.X/ W S.X/! S.X/ be chain maps, natural in X, which coincide on S0.X/. Then there exists a natural chain homotopy from a0 to a1. This is a consequence of (11.5.1) for F D G and the models n as in the Þ proof of (9.3.3). Problems 1. Let n W n! n,.0; : : : ; n/ 7!.n; : : : ; 0/. Verify that Sn.X/! Sn.X/, 7!.1/.nC1/n=2n is a natural chain map. By (9.3.5), it is naturally homotopic to the identity. 2. One can prove the homotopy invariance by constructing an explicit chain homotopy. A natural construction would associate to a singular n-simplex W n
! X the singular prism id W n I! X I. The combinatorial (set-theoretic) boundary of n I is n 1 [ n 0 [.@n/ I, and this corresponds exactly to the deﬁnition of a chain homotopy, if one takes orientations into account. This idea works; one has to decompose n I into simplices, and it sufﬁces to do this algebraically. In the prism n I let 0; 1; : : : ; n denote the vertices of the base and 00; 10; : : : ; n0 those of the top. In the notation for afﬁne singular simplices introduced later, show that an explicit formula for a D snn is P snn D n iD0.1/i Œ0; 1; : : : ; i; i 0;.i C 1/0; : : : ; n0: (This is a special case of the Eilenberg–Mac Lane shufﬂe morphism to be discussed in the section on homology products.) 9.4 Barycentric Subdivision. Excision 9.4. Barycentric Subdivision. Excision 231 The basic property of homology is the excision theorem (9.4.7). It is this theorem which allows for effective computations. Its proof is based on subdivision of standard simplices. We have to work out the algebraic form of this subdivision ﬁrst. Let D Rn be convex, and let v0; : : : ; vp be elements in D. The afﬁne singular i i vi will be denoted D Œv0; : : : ; vp. With i i ei 7! P P simplex W p! D, this notation @Œv0; : : : ; vp D P p iD0.1/i Œv0; : : : ; bvi ; : : : ; vp; where bvi means that vi has to be omitted from the string of vertices. For each v 2 D we have the contracting homotopy D I! D,.x; t/ 7!.1 t/x C tv. If we apply the cone construction 9.3.1 to Œv0; : : : ; vp we obtain Œv; v0; : : :
; vp. We denote the chain homotopy associated to the contraction by Sp.D/! SpC1.D/, c 7! v c. We have forc 2 Sp.D/: (.1/ @.v c/ D c v @c; p > 0; c ".c/v; p D 0; P P with " W S0.D/! Z, n 7! The barycenter of D Œv0; : : : ; vp is ˇ D 1 pC1 n. P p iD0 vi. We deﬁne induc- tively Bp.X/ D Bp W Sp.X/! Sp.X/ to be the homomorphism which sends W p! X to Bp./ D #Bp.p/, where Bp.p/ is deﬁned inductively as (.2/ Bp.p/ D 0; p D 0; ˇ p Bp1.@p/; p > 0: (9.4.1) Proposition. The Bp constitute a natural chain map which is naturally homotopic to the identity. Proof. The equalities f#B D f##B.p/ D.f /#B.p/ D B.f / D Bf# prove the naturality. We verify by induction over p that we have a chain map. Let p D 1. Then @B.1/ D @.ˇ 1 B.@1// D @1 D B@.1/. For p > 1 we compute p B@@p D B@p: p B.@p// D B@p ˇ p @B@p D B@p ˇ @Bp D @.ˇ We have used: Deﬁnition; (1); inductive assumption; @@ D 0. We now use this special case and the naturality B@ D B@#p D B#@p D #B@p D #@Bp D @#Bp D @B; 232 Chapter 9. Singular Homology and this computation covers the general case. The chain map B is naturally homotopic to the identity (see (9.3.5)). Let U be a family of subsets of X such that their interiors cover X. We call a singular simplex U-small, if its image is
contained in some member of U. The subgroup spanned by the U-small simplices is a subcomplex S U.X/ of S.X/ with homology groups denoted by H U n.X/. (9.4.2) Lemma. The diameter d.v0; : : : ; vp/ of the afﬁne simplex Œv0; : : : ; vp with respect to the Euclidean norm is the maximum of the kvi vj k. P P Proof. Let x; y 2 Œv0; : : : ; vp and x D j vj. Then, because of j D 1, P kx yk D k j.vj y/k P j kvj yk max j kvj yk: This shows in particular ky vi k max j kvj vi k; we insert this in the above and obtain kx yk max i;j kvi vj k; hence the diameter is at most as stated. On the other hand, this value is clearly attained as the distance between two points. (9.4.3) Lemma. Let v0; : : : ; vp 2 Rn. Then BpŒv0; : : : ; vp is a linear combination of afﬁne simplices with diameter at most p pC1 d.v0; : : : ; vp/. Proof. From the inductive deﬁnition (2) and the naturality of B we conclude.3/ BŒv0; : : : ; vp D P p j D0.1/j ˇ BŒv0; : : : ; bvj ; : : : ; vp where D Œv0; : : : ; vp. We prove the claim by induction over p. The assertion is obvious for p D 0, a point has diameter zero. By induction hypothesis, the simplices in the chain BŒv0; : : : ; vj ; : : : ; vp are afﬁne of diameter at most p1 p d.v0; : : : ; vj ; : : : ; vp/ p1 p d.v0; : : : ; vp/. The simplices in BŒv0; : : : ; vp have vert
ices ˇ and vertices from simplices in BŒv0; : : : ; vj ; : : : ; vp. It sufﬁces to evaluate the distance of ˇ from such vertices. It is less than or equal to sup.k ˇ xk jx 2 Œv0; : : : ; vp/. j vj. Then k ˇ xk max k ˇ vj k, as in the proof of (9.4.2). Let x D Moreover we have P k ˇ vj k D P 1 1 pC1 maxi;j kvi vj k D p vj i vi pC1 p i kvi vj k pC1 pC1 d.v0; : : : ; vp/: P Since.p 1/=p < p=.p C 1/ we have veriﬁed, altogether, the claim. (9.4.4) Lemma. Let W p! X be a singular simplex. Then there exists a k 2 N such that each simplex in the chain Bk has an image contained in a member of U. (Here Bk is the k-fold iteration of B.) 9.4. Barycentric Subdivision. Excision 233 Proof. We consider the open covering. 1.U ı//; U 2 U of p. Let " > 0 be a Lebesgue number of this covering. The simplices of Bk arise by an application of to the simplices in Bkp. From (9.4.3) we see that the diameter of these simplices is at most. p pC1 /kd.e0; : : : ; ep/. If k is large enough, this number is smaller than "..X/! H.X/..X/ S.X/ induces an (9.4.5) Theorem. The inclusion of chain complexes S U isomorphism H U Proof. Let a 2 S U n.X/ be a cycle which represents a homology class in the kernel. Thus a D @b with some b 2 SnC1.X/. By (9.4.4), there exists k such that Bk.b/ 2 S U nC1.X/ (apply (9.4.4) to the ﬁnite number of simpl
ices in the linear combination of b). By (9.4.1), there exists a natural chain homotopy Tk between Bk and the identity. Therefore Bk.b/ b D Tk.@b/ C @Tk.b/ D Tk.a/ C @Tk.b/; and we conclude @Bk.b/ @b D @Tk.a/; From the naturality of Tk and the inclusion a 2 S U Therefore a is a boundary in S U question. nC1.X/..X/. This shows the injectivity of the map in n.X/ we see Tk.a/ 2 S U a D @b D @.Bk.b/ Tk.a//: Let a 2 Sn.X/ be a cycle. By (9.4.4), there exists k such that Bka 2 S U n.X/. We know that Bka a D Tk.@a/ C @Tk.a/ D @Tk.a/: Since Bk is a chain map, Bka is a cycle. From the last equality we see that a is homologous to a cycle in S U n.X/. This shows the surjectivity of the map in question. deﬁne the chain complex S U H U Let now.X; A/ be a pair of spaces. We write U \ A D.U \ A j U 2 U/ and.A/ with homology groups.X; A/. We obtain a commutative diagram of chain complexes with exact rows: S U.X; A/ D S U.X/=S U\A S U\A.A/ S U.X; A/ 0 0 0 S.A/ S.X/ S.X; A/ 0. Each row has its long exact homology sequence. We apply (9.4.5) to.X; U/ and.A; U \ A/, use the Five Lemma (11.2.7), and obtain: (9.4.6) Theorem. The inclusion of chain complexes W S U duces an isomorphism H U see that the inclusion is actually a chain equivalence..X; A/! S.X; A/ in.X; A/ Š H.X; A/. By an application of (11.6.3)
we 234 Chapter 9. Singular Homology 1 [ Y ı (9.4.7) Theorem (Excision Theorem). Let Y D Y ı 2. Then the inclusion induces an isomorphism H.Y2; Y1 \ Y2/ Š H.Y; Y1/. Let B A X and suppose that xB Aı. Then the inclusion.X X B; A X B/!.X; A/ induces an isomorphism H.X X B; A X B/ Š H.X; A/. Again we can invoke (11.6.3) and conclude that the inclusion actually induces chain equivalences between the chain complexes under consideration. Proof. The covering U D.Y1; Y2/ satisﬁes the hypothesis of (9.4.5). By deﬁnition, we have S U n.X/ D Sn.Y1/ C Sn.Y2/ and also Sn.Y1 \ Y2/ D Sn.Y1/ \ Sn.Y2/. The inclusion S.Y2/! S.Y / induces therefore, by an isomorphism theorem of elementary algebra, Sn.Y2/ Sn.Y1 \ Y2/ D Sn.Y2/ Sn.Y1/ \ Sn.Y2/ Š Sn.Y1/ C Sn.Y2/ Sn.Y1/ D S U n.Y / Sn.Y1/ : By (9.4.5) and (11.2.7) we see, ﬁrstly, that S U.Y /=S.Y1/! S.Y /=S.Y1/ and, altogether, that S.Y2/=S.Y1 \ Y2/! S.Y /=S.Y1/ induces an isomorphism in homology. The second statement is equivalent to the ﬁrst; we use X D Y, A D Y1, X X B D Y2. Problems 1. Let D Rm and E Rn be convex and let f W D! E be the restriction of a linear map. Then f#.v c/ D f.v/ f#.c/. 2. Although not necessary for further investigations, it might be interesting to describe the chain BŒv0; : : : ; vp in detail. We use (
3) in the proof of (9.4.3). By (2), formula (3) also holds for Œv0; : : : ; vp. This yields BŒv0; v1 D Œv01; v1 Œv01; v0 with barycenter v01, and for BŒv0; v1; v2 we obtain in short-hand notation what is illustrated by the next ﬁgure. Œ012; 12; 2 Œ012; 12; 1 Œ012; 02; 2 C Œ012; 02; 0 C Œ012; 01; 1 Œ012; 01; 0: 3 v1 C 1 3 v2 Œ012; 12; 2 1 v2 3 v0 C 1 1 2 v1 C 1 C C C 2 v2 v1 v0 One continues inductively in this manner. Let S.p C 1/ denote the permutation group of f0; : : : ; pg. We associate to D Œv0; : : : ; vp and 2 S.p C 1/ the simplex D Œv r D Œv.r/; : : : ; v.p/ˇ. With this notation the following holds: B D p, where v 2S.pC1/ sign./. 0 ; : : : ; v P 9.5. Weak Equivalences and Homology 235 9.5 Weak Equivalences and Homology Although singular homology groups are deﬁned for arbitrary topological spaces, they only capture combinatorial information. The theory is determined by its values on cell complexes. Technically this uses two facts: (1) a weak homotopy equivalence induces isomorphisms of homology groups; (2) every topological space is weakly equivalent to a CW-complex. One can use cell complexes to give proofs by induction over the skeleta. Usually the situation for a single cell is quite transparent, and this fact makes the inductive proofs easy to follow and to remember. Once a theorem is known for cell complexes, it can formally be extended to general topological spaces. We now prove this invariance property of singular homology [56], [21]. Let.X; A; / be a pointed pair. Let Œkn be the n-skeleton of the standard simplicial complex Œk (this is the reason for
switching the notation for the standard k-simplex). Let S.n;A/.X/ for n 0 denote the subgroup of Sk.X/ spanned by the singular simplices W Œk! X with the property k.#/ The groups.S.n;A/ S.X/. k.Œkn/ A:.X/ j k 0/ form the Eilenberg subcomplex S.n;A/.X/ of (9.5.1) Theorem. Let.X; A/ be n-connected. Then the inclusion of the Eilenberg subcomplex ˛ W S.n;A/.X/! S.X/ is a chain equivalence. Proof. We assign to a simplex W Œk! X a homotopy P./ W Œk I! X such that (1) P./0 D, (2) P./1 satisﬁes (#), (3) P./t D, provided satisﬁes already (#), i id/ D P. ı d k (4) P./ ı.d k i /. According to (3), the assignment P is deﬁned for simplices which satisfy (#). For the remaining simplices we use an inductive construction. Suppose k D 0. Then.Œ0/ 2 X is a point. Since.X; A/ is 0-connected, there exists a path from this point to a point in A. We choose a path of this type as P./. Suppose P is given for j -simplices, j < k. Then for each k-simplex the i / combine to a homotopy homotopy P. ı d k @Œk I! X. Moreover P./0 is given. Altogether we obtain i / is already deﬁned, and the P. ı d k QP./ W.Œk 0 [ @Œk I; @Œk 1/!.X; A/: Let k n. Then Œkn D Œk, and similarly for the faces. By the inductive assumption, QP./ sends @Œk 1 into A. 236 Chapter 9. Singular Homology There exists a homeomorphism W Œk I! Œk I which induces home- omorphisms (see (2.3.
6)) Œk 0 Š Œk 0 [ @Œk I; @Œk 0 Š @Œk 1; @Œk I [ Œk 1 Š Œk 1: Since.X; A/ is k-connected, the map QP./ ı W.Œk 0; @Œk 0/!.X; A/ can be extended to a homotopy Q W Œk I! X which is constant on @Œk I and sends Œk 1 into A. We now set P./ D Q ı 1. Then P./ extends QP./, hence (1) and (4) are satisﬁed, and (2) also holds by construction. Let k > n. We use the coﬁbration.Œk; @Œk/ in order to extend QP./ to P./. Since Œkn @Œk, we see that P./1 satisﬁes (#). We now deﬁne W Sk.X/! S.n;A/.X/ by 7! P./1. Property (4) shows that is a chain map, and ı ˛ D id holds by construction. We deﬁne s W Sk.X/! SkC1.X/ by s./ D P./#h.k/ k k 2 Sk.Œk/ h SkC1.Œk I / P./# SkC1.X/ 3 s./ where h is the natural chain homotopy between i 0 tations # and i 1 #, see (9.3.3). The compu- @s./ D @.P./#h.k// D P./#@h.k/ D P./1#.k/ P./0#.k/ P./#h.@k/ D./ P./#h.@k/; P P s@./ D s.1/i ı d k i.1/i d k P D D P./#.1/i P. ı d k i /#h.k1/ i#h.k1/ D P./#h.@k/ show that s is a chain homotopy between ˛ ı and id. For k n we have Œkn
D Œk and therefore S.n;A/.X/ D Sk.A/. The chain equivalence (9.5.1) and the exact homology sequence of.X; A/ now yield: k (9.5.2) Theorem. Let.X; A/ be n-connected. Then Hk.A/ Hk.X; A/ D 0 for k n. Š! Hk.X/ and Let f W X! Y be a weak homotopy equivalence. We can assume that f is an inclusion (mapping cylinder and homotopy invariance). 9.6. Homology with Coefﬁcients 237 (9.5.3) Theorem. A weak homotopy equivalence induces isomorphisms of the sin- gular homology groups. (9.5.4) Remark. Suppose that.X; A; / is a pointed pair and A is pathwise connected. Then we can deﬁne a subcomplex S.X;A;/.X/ of S.X/ where we require in addition to (#) that.Œk0/ D fg. Again the inclusion is a chain equivalence. Þ 9.6 Homology with Coefﬁcients Let C D.Cn; cn/ be a chain complex of abelian groups and let G be a further abelian group. Then the groups Cn ˝ G and the boundary operators cn ˝ id form again a chain complex (the tensor product is taken over Z). We denote it by C ˝G. We apply this process to the singular complex S.X; A/ and obtain the complex S.X; A/ ˝ G of singular chains with coefﬁcients in G. Its homology group in dimension n is denoted Hn.X; AI G/. The cases G D Z; Q; Z=p are often referred to as integral, rational, mod.p/ homology. Chains in Sn.X; A/ ˝ G can a, a 2 G of singular nbe written as ﬁnite formal linear combinations simplices ; this accounts for the name “chain with coefﬁcients”. The sequence 0! S.A/! S.X/! S.X; A/! 0 remains exact when tens
ored with G, i.e., Sn.X; A/ ˝ G Š Sn.X/ ˝ G=Sn.A/ ˝ G. Therefore we still have the exact homology sequence (11.3.2) P! Hn.AI G/! Hn.XI G/! Hn.X; AI G/ @! Hn1.AI G/! and the analogous sequence for triples. The boundary operators @ are again natural transformations. If 0! G0! G! G00! G is an exact sequence of abelian groups, then the tensor product with S.X; A/ yields again an exact sequence of chain complexes and we obtain from (11.3.2) an exact sequence of the form! Hn.X; AI G0/! Hn.X; AI G/! Hn.X; AI G00/! Hn1.X; AI G0/! : The passage from C to C˝G is compatible with chain maps and chain homotopies. A chain equivalence induces a chain equivalence. This fact yields the homotopy invariance of the homology groups Hn.X; AI G/. The excision theorem still holds. This is a consequence of (9.4.7): Under the hypothesis of the excision theorem, the chain equivalence S.Y1; Y1 \ Y2/! S.Y; Y2/ induces a chain equivalence when tensored with G. Hence the functors Hn.X; AI G/ satisfy the axioms of Eilenberg and Steenrod for a homology theory. The dimension axiom holds: We have a canonical isomorphism "P W H0.P / Š G for a point P, which maps the homology class of the chain a to a, where is the unique 0-simplex. The application of (11.9.1) to topology uses the fact that the singular chain complex consists of free abelian groups. Therefore we obtain: 238 Chapter 9. Singular Homology (9.6.1) Theorem (Universal Coefﬁcients). Let R be a principal ideal domain and G an R-module. There exists an exact sequence 0! Hn.X; AI R/ ˝R G ˛! Hn.X; AI G/! Tor
.Hn1.X; AI R/; G/! 0: The sequence is natural in.X; A/ and G. The sequence splits, the splitting is natural in G, but not in.X; A/. The splitting statement means that Hn.X; AI G/ can be determined as an abelian group from homology with coefﬁcients in Z, but the functor Hn.I G/ is not the direct sum of the functors Hn.I Z/ ˝ G and Tor.Hn1.I Z/; G/. Here If f W.X; A/!.Y; B/ induces an isomorphism is a consequence of (9.6.1): f W H.X; A/ Š H.Y; B/ for Dn 1; n, then it induces also an isomorphism Hn.X; AI G/ Š Hn.Y; BI G/. 9.7 The Theorem of Eilenberg and Zilber We study the homology of products. For this purpose we compare the chain complexes S.X/ ˝ S.Y / and S.X Y /. Both are values at.X; Y / of a functor TOP TOP! CHC into the category of chain complexes which are zero in negative degrees. In dimension zero they essentially coincide. For x 2 X let x 2 S0.X/ also denote the basis element given by the singular simplex 0! fxg X. Then S.X Y / has the basis.x; y/ and S.X/ ˝ S.Y / the basis x ˝ y for.x; y/ 2 X Y. Natural transformations P W S./ ˝ S./! S. / and Q W S. /! S./ ˝ S./ are called an Eilenberg–Zilber morphisms if in dimension zero always P.x ˝ y/ D.x; y/ and Q.x; y/ D x ˝ y. Both functors are free and acyclic in the sense of (11.5.1). For.S./ ˝ S.//n we use the models.k; nk/ and the elements id ˝ id; for Sn. / we use the models.n; n/ and the diagonal maps n! n n.
They account for the freeness. The homology of the chain complexes S.p q/ is zero in positive dimensions, since p q is contractible; the homology of S.p/˝S.q/ is zero in positive dimensions, since the tensor product of chain complexes is compatible with chain homotopies, and the chain complexes S.p/ are homotopy equivalent to the trivial complex. Similar statements hold for the analogous functors in three (or more) variables like S.X Y Z/ or the corresponding three-fold tensor products. As an application of (11.5.1) we obtain: (9.7.1) Theorem. (1) Eilenberg–Zilber morphisms P and Q exist. For each pair.P; Q/ of Eilenberg–Zilber morphisms the compositions P ı Q and Q ı P are naturally homotopic to the identity. Hence the PX;Y and QX;Y are chain equivalences and any two Eilenberg–Zilber morphisms P; P 0 are naturally homotopic (similarly for Q; Q0). (2) An Eilenberg–Zilber morphism P is associative and commutative up to natural homotopy, i.e., the natural transformations PXY;Z ı.PX;Y ˝ 1/ and 9.7. The Theorem of Eilenberg and Zilber 239 PX;Y Z ı.1 ˝ PY;Z/ from S.X/ ˝ S.Y / ˝ S.Z/ to S.X Y X/ are naturally homotopic and the transformations.tX;Y /# ı PX;Y and PY;X ı X;Y are naturally homotopic. Here tX;Y W X Y! Y X interchanges the factors and X;Y.x ˝ y/ D.1/jxjjyjy ˝ x. (3) An Eilenberg–Zilber morphism Q is coassociative and cocommutative up to natural homotopy, i.e., the natural transformations.QX;Y ˝ 1/ ı QXY;Z and.1˝QY;Z/ıQX;Y Z are naturally homotopic
, and the transformations X;Y ıQX;Y and QY;X ı.tX;Y /# are naturally homotopic. As a consequence one can determine the homology of X Y from the chain complex S.X/ ˝ S.Y /. We now turn to relative chain complexes and abbreviate S D S. (9.7.2) Proposition. For Eilenberg–Zilber transformations P; Q and pairs of spaces.X; A/;.Y; B/ we have a commutative diagram with short exact rows S.A/ ˝ S.Y / C S.X/ ˝ S.B/ S.X/ ˝ S.Y / S.X; A/ ˝ S.Y; B/ Q0 P 0 Q P Q00 P 00 S.A Y / C S.X B/ S.X Y / S.X Y / S.A Y / C S.X B/ : The vertical maps are induced by P and Q. The compositions P 0Q0, Q0P 0, P 00Q00, Q00P 00 are naturally homotopic to the identity. Proof. The naturality of P shows P.S.A/ ˝ S.Y // S.A Y / and similarly for Q. This shows that P; Q induce by restriction P 0; Q0, and P 00; Q00 are the homomorphisms induced on the quotients. Since the homotopy PQ'id is natural, it maps S.A Y / C S.X B/ into itself and shows P 0Q0'id. 9.7.3 We can compose P W S.X; A/ ˝ S.Y; B/! S.X Y /=.S.A Y / C S.X B// with the map induced by the inclusion S.A Y / C S.X B/ S.A Y [ X B/ and obtain altogether natural chain maps P W S.X; A/ ˝ S.Y; B/! S..X; A/.Y; B//: We call the pair.A Y; X B/ excisive, if this chain map is a chain equivalence. Þ 240 Chapter 9. Singular Homology 9.7.4 The natural chain map P induces natural chain maps for singular chain
complexes with coefﬁcients. Let R be a commutative ring and M; N R-modules. S.X; AI M / ˝ S.Y; BI N / D.S.X; A/ ˝ M / ˝R.S.Y; B/ ˝ N /!.S.X; A/ ˝ S.Y; B// ˝.M ˝R N /!.S..X; A/.Y; B/// ˝.M ˝R N /: In many cases this chain map is followed by a homomorphism induced by a linear map M ˝R N! L. Examples are R ˝R R! R, x ˝ y 7! xy in the case of a Þ ring R and R ˝R N! N, x ˝ n 7! x n in the case of an R-module N. Problems 1. There exist explicit Eilenberg–Zilber morphisms which have further properties. Let p; q 2 N. We use the notation Œn D f0; 1; : : : ; ng. A.p; q/-shufﬂe is a map W Œp Cq! ŒpŒq with.0/ D.0; 0/ and.p C q/ D.p; q/ such that both components of D.1; 2/ are (weakly) increasing. Given, there exists a permutation.; / D.1; : : : ; p; 1; : : : ; q/ of 1; 2; : : : ; p C q such that 1 1 < < p p C q; 1 1; : : : ; q p C q and 1.j / > 1.j 1/ and 2.k/ > 2.k 1/. We denote the signum of the permutation.; / by "./. If we interpret the points.0/; : : : ;.p C q/ in the integral lattice Œp Œq as the vertices of an edge-path from.0; 0/ to.p; q/, then the step.j /!.j C 1/ is horizontal or vertical of length 1. In the convex set p q we have the afﬁne.p C q
/-simplex Œ.0/; : : : ;.p C q/, also denoted, and the set of these simplices form a triangulation of the product when runs through the.p; q/-shufﬂes †.p; q/. (We do not need this geometric fact, but it explains the idea of the construction.) Deﬁne P s p;q W Sp.X/ ˝ Sq.Y /! SpCq.X; Y /; ˝ 7! P 2†.p;q/ "./.. / ı / on a pair ; of singular simplices. The P s p;q are a strictly associative Eilenberg–Zilber morphism. We call it the shufﬂe morphism or the Eilenberg–Mac Lane morphism. 2. An approximation of the diagonal is a natural chain map D W S.X/! S.X/ ˝ S.X/ which coincides in dimension zero with x 7! x ˝ x. (The name refers to the fact that the diagonal of a cellular complex is not a cellular map, and so one looks for a homotopic cellular approximation.) By an application of (11.5.1) one shows that any two approximations of the diagonal are naturally chain homotopic. 3. The classical approximation of the diagonal is the Alexander–Whitney map. Let W n! X be an n-simplex, n D p C q, 0 p; q n. We have the afﬁne maps ap W p! n, ei 7! ei and bq W q! n, ei 7! enqCi. They are used to deﬁne p D ı ap and 2 1 q D ı bq. The Alexander–Whitney approximation of the diagonal is deﬁned by Dn D P pCqDn.8. The Homology Product 241 and linear extension. 4. Given an approximation D of the diagonal one constructs from it an Eilenberg–Zilber morphism Q as the composition S.X Y / DXY S.X Y / ˝ S.X Y / prX # ˝ prY # S.X/ ˝ S.Y /: Let QAW be the Eilenberg–Zilber morphism obtained
from the Alexander–Whitney approximation of the diagonal and call it the Alexander–Whitney morphism. The Alexander– Whitney morphism is strictly coassociative. 5. The Eilenberg–Mac Lane morphism EM and the Alexander–Whitney morphism AW are also compatible in a certain sense: S.W X/ ˝ S.Y Z/ EM S.W X Y Z/ AW ˝AW.1tX;Y 1/# S.W / ˝ S.X/ ˝ S.Y / ˝ S.Z/ S.W Y X Z/ 1˝X;Y ˝1 AW S.W / ˝ S.Y / ˝ S.X/ ˝ S.Z/ EM ˝EM S.W Y / ˝ S.X Z/ commutes. 9.8 The Homology Product We pass to homology from the chain map P in (9.7.4) H.X; AI M / ˝ H.Y; BI N / D H.S.X; AI M // ˝ H.S.Y; BI N //! H.S.X; AI M / ˝ S.Y; BI N //! H.S..X; A/.Y; B//I M ˝ N / These maps are natural transformations, and we call them the homology product. We use the notation x ˝ y 7! x y for the homology product. In the case of M D N D R we combine with the map induced by the canonical isomorphism R ˝R R! R and obtain a homology product Hi.X; AI R/ ˝ Hj.Y; BI R/! HiCj..X; A/.Y; B/I R/: In general we can compose with a bilinear map M ˝N! P ; for instance we can use an R-module structure R ˝ M! M on M. We list some formal properties of the homology product, for simplicity of notation only for homology with coefﬁcients in R. We use the following notation: f W.X; A/!.X 0; A0/ and g W.Y; B/!.Y 0; B 0/ are continuous maps. Let.X B; A

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