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zero. The sections correspond to maps f W S n! Rn such that f.x/ D f.x/. One form of the theorem of Borsuk–Ulam says that maps of this type always have a zero. We would reprove this result, if we show that the Euler class mod (2) is non-zero. The tautological bundle over RP n has as Euler class the non-zero element w of H 1.RP nI Z=2/. The Euler classes are multiplicative and D n. Hence e./ D wn 6D 0. 18.9 Euler Class and Euler Characteristic Let M be a closed orientable n-manifold. We deﬁne in a new manner the Thom class of the tangent bundle. It is an element t.M / 2 H n.M M; M M X D/ such that for each x 2 M the restriction of t.M / along H n.M M; M M X D/! H n.x M; x.M X x// is a generator (integral coefﬁcients, D the diagonal). The image of t.M / under the composition H n.M M; M M X D/! H n.M M / d! H n.M / (where d is the diagonal map) is now called the associated Euler class e.M / of M. Let us use coefﬁcients in a ﬁeld K. We still denote the image of the fundamental class ŒM 2 Hn.M I Z/ in Hn.M I K/ by ŒM. We use the product orientation ŒM M D ŒM ŒM. Let W E./! M be the normal bundle of the diagonal d W M! M M with disk- and sphere bundle D./ and S./ and tubular map j W D./! M M. The fundamental class of ŒM M 2 H2n.M M / induces a fundamental class ŒD./ 2 H2n.D./; S.// via H2n.M M /! H2n.M M; M M X D/ Š H2n.D./; S.//: 18.9. Euler Class and Euler Characteristic 465 Let z D j.ŒD./. The diagram H n.M M / H
n.M M; M M X D/ Z ŒM M Z z Hn.M M / D Hn.M M / j Š j H n.D./; S.// Z ŒD./ Hn.D.// commutes (naturality of the cap product). Suppose M is connected. From the isomorphisms Hn.M / i Š Hn.D.// Z ŒD./ Š H n.D./; S.// we obtain an element t./ that satisﬁes iŒM D t./ Z ŒD./. It is a generator and therefore a Thom class. We deﬁne t.M / 2 H n.M M; M M X D/ by j t.M / D t./. The image.M / 2 H n.M M / of t.M / is characterized by the relation Z ŒM M D dŒM. From the deﬁnitions we see that d is the i image of t./ under H n.D./; S.//! H n.D.//! H n.M /, hence the Euler class e./ of. Let B D f˛g be a basis of H.M / and f˛0g the dual basis in H.M / with respect to the intersection form h˛0 Y ˇ; ŒM i Dı ˛ˇ, j˛0j Dn j˛j. (18.9.1) Proposition. The image.M / 2 H n.M M / of t.M / is given by P ˛2B.1/j˛j˛0 ˛ 2 H n.M M /: D A consequence is e.M / D d D P he.M /; ŒM i D P ˛.1/h˛0 Y ˛; ŒM i D ˛.1/j˛j D.M /: ˛2B.1/j˛j˛0 Y ˛; P Proof. The Künneth isomorphism H.M / ˝ H.M / Š H.M M /; u ˝ v 7! u v; ;ı2B A.; ı/ 0
ı. The tells us that there exists a relation of the form D following computations determine the coefﬁcient A.; ı/. Let ˛ and ˇ be basis elements of degree p. Then P h.˛ ˇ0/ Y ; ŒM M i D h˛ ˇ0; Z ŒM M i D h˛ ˇ0; dŒM i D hd.˛ ˇ0/; ŒM i D h˛ Y ˇ0; ŒM i D.1/p.np/hˇ0 Y ˛; ŒM i D.1/ p.np/ı˛ˇ : 466 Chapter 18. Duality A second computation gives h.˛ ˇ0/ Y ; ŒM M i D h.˛ ˇ0/ Y P P P A.; ı/ 0 ı; ŒM M i A.; ı/.1/jˇ 0jj 0jh.˛ Y 0/.ˇ0 Y ı/; ŒM ŒM i A.; ı/.1/jˇ 0jj 0j.1/n.jˇ 0jCjıj/h˛ Y 0; ŒM ihˇ0 Y ı; ŒM i: D D Only summands with D ˛ and ı D ˇ are non-zero. Thus this evaluation has the value A.˛; ˇ/ D.1/pn (collect the signs and compute modulo 2). We compare the two results and obtain A.˛; ˇ/ D.1/pı˛ˇ. Chapter 19 Characteristic Classes Characteristic classes are cohomological invariants of bundles which are compatible with bundle maps. Let h./ be a cohomology theory. An hk-valued characteristic class for numerable n-dimensional complex vector bundles, say, assigns to each such bundle W E./! B an element c./ 2 hk.B/ such that for a bundle map! over f W B! C the naturality property f c./ D c./ holds. An assignment which has these properties is determined by its value c.n/ 2 hk.BU.n// on the universal bundle n, and this value can
be prescribed in an In other words, the elements of hk.BU.n// arbitrary manner (Yoneda lemma). correspond to this type of characteristic classes. It turns out that in important cases characteristic classes are generated by a few of them with distinguished properties, essentially a set of generators of the cohomology of classifying spaces. We work with a multiplicative and additive cohomology theory h and bundles are assumed to be numerable. A C-orientation of the theory assigns to each n-dimensional complex vector bundle (numerable, over a CW-complex,…) W E./! B a Thom class t./ 2 h2n.E./; E0.// such that for a bundle map f W E./! E./ the naturality f t./ D t./ holds and the Thom classes are multiplicative t./ t./ D t. /. If an assignment of this type is given, then the theory is called C-oriented. In a similar manner we call a theory R-oriented, if for each ndimensional real vector bundle W E./! B a Thom class t./ 2 hn.E./; E0.// is given which is natural and multiplicative. It is a remarkable fact that structures of this type are determined by 1-dimensional bundles. (19.0.1) Theorem. A C-orientation is determined by its value t.1/ 2 h2.E.1/; E0.1// on the universal 1-dimensional bundle 1 over CP 1. Each Thom class t of 1 determines a C-orientation. A similar bijection exists between Thom classes of the universal 1-dimensional real vector bundle over RP 1 and R-orientations. An example of a C-oriented theory is H.I Z/; a complex vector bundle has a canonical Thom class and these Thom classes are natural and multiplicative. One can use an arbitrary commutative ring as coefﬁcient ring. An example of an R-oriented theory is H.I Z=2/; a real vector bundle has a unique Thom class in this theory. One can use any commutative ring of characteristic 2 as coefﬁcient ring. 468 Chapter 19. Characteristic Classes Suppose the theory is C-oriented. Then an n-dimensional complex vector bundle over B has an Euler class e./ 2 h2n.B/ associated to t./. Euler classes are natural, f
e./ D e.f /, and multiplicative, e. ˚ / D e./ Y e./. The Thom classes have associated Thom homomorphisms. They are deﬁned as before by cup product with the Thom class ˆ./ W hk.B; A/! hkC2n.E./; E./0 [ E.jA//; x 7!.x/ Y t./: These Thom homomorphisms are natural and multiplicative as we have explained earlier. For an R-oriented theory we have natural and multiplicative Euler classes for real vector bundles. A proof of (19.0.1) is based on a determination of characteristic classes. We present a construction of characteristic classes based on the cohomology of projective bundles. For this purpose, classifying spaces are not used. But they will of course appear and they are necessary for a more global view-point. 19.1 Projective Spaces Let n W En! CP n1 be the canonical bundle with total space En D Cn X 0 C C;.z; u/.z; u/: We have the embedding n W En! CP n,.z; u/ 7! Œz; u. The image is the complement of the point DŒ0; : : : ; 0; 1. Let t.n/ 2 h2.En; E0 n/ be a Thom class. The Thom class yields the element tn 2 h2.CP n/ as the image under h2.En; E0 n/ Š h2.CP n; CP n X CP n1/ Š h2.CP n; /! h2.CP n/: The ﬁrst isomorphism is induced by n. Note that n sends the zero section to CP n1, the image under the embedding W Œx1; : : : ; xn 7! Œx1; : : : ; xn; 0. The total space En of n was denoted H.1/ in (14.2.6). The bundle n is the (complex) normal bundle of the embedding W CP n1! CP n. The embedding n is a tubular map; it also shows that CP n is the one-point compactiﬁcation of En (see the deﬁnition of a Thom space in the �
�nal chapter). The complement CP n X CP n1 is the afﬁne subset UnC1 D fŒx1; : : : ; xnC1 j xnC1 6D 0g. We obtain a homomorphism hŒtn! h.CP n/ of graded h-algebras; it sends t nC1 to zero (see (17.2.5)) and induces a homomorphism of the quotient by the principal ideal.t nC1 /. n n (19.1.1) Lemma. Let t.n1/ 2 h2.En1; E0 n1/ be the Thom class obtained from t.n/ by restriction along. Let tn1 2 h2.CP n1/ be obtained from t.n1/ as explained above. Then: 19.1. Projective Spaces 469 (1) tn D tn1. (2) tn1 is the Euler class associated to t.n/. Proof. (1) The embedding is homotopic to the embedding 1 W Œx1; : : : ; xn 7! Œ0; x1; : : : ; xn. Thus it sufﬁces to show 1tn D tn1. We have a bundle map 2 W En1! En which is compatible with the embeddings, i.e., n2 D 1n1. We apply cohomology to this commutativity and obtain the desired result. (2) With the zero section s the diagram h2.CP n; CP n X CP n1/ h2.CP n/ Š n h2.En; E0 n/ s h2.CP n1/ commutes. The deﬁnition of the Euler class and (1) now yield the result. (19.1.2) Lemma. For singular homology and cohomology the Kronecker pairing relation ht1; ŒCP 1i D1 holds. The element t1 is by (19.1.1) also the Euler class of 2 and this is, by deﬁnition, the ﬁrst Chern class. Proof. 1 is a bundle over a point. We have the isomorphism'W C! E1, z 7!.1; z/. By de
ﬁnition of the canonical Thom class of a complex vector bundle the Thom class t.1/ 2 H 2.E1; E0 1 / is mapped to the generator e.2/ under ', where e.2/ is deﬁned by the relation he.2/; e2 i D1 (Kronecker pairing). The element t1 is the image of e.2/ under H 2.C; C X 0/! H 2.CP 1; CP 1 X CP 0/! H 2.CP 1/ and the fundamental class ŒCP 1 is mapped to e2 under H2.CP 1/! H2.CP 1; CP 1 X CP 0/! H2.C; C X 0/. Naturality of the Kronecker pairing now gives the desired result. (19.1.3) Theorem. The homomorphism just constructed is an isomorphism h.CP n/ Š hŒtn=.t nC1 / of graded h-algebras. In particular h.CP n/ is a free h-module with basis 1; tn Proof. Induction on n. We have the Thom isomorphism hk.CP n1/! hkC2.En; E0 n/; By induction, h.CP n1/ is a free h-module with basis 1; tn1; : : : ; t n1 therefore h.En; E0 nx [ t.n/ D x t.n/: x 7! n1 and n/ is a free module with basis 1 t.n/; tn1 t.n/; : : : ; t n1 n1 t.n/: n/ Š h.CP n; / constructed above and claim We apply the isomorphism h.En; E0 that it sends t k. Unraveling the deﬁnitions one shows that this claim is a consequence of the naturality of the cup product and the fact (19.1.1) that tn1 is the Euler class of t.n/. n1 t.n/ to t kC1 n 470 Chapter 19. Characteristic Classes We denote by hŒŒT the ring of homogeneous formal power series in T over the graded ring h. If T has the degree 2, then the power series in hŒ�
�T of degree j aj T j with aj 2 hk2j. If h is concentrated in degree k consist of the series zero, then this coincides with the polynomial ring h0ŒT. P Let t.1/ 2 h2.E1; E0 1/ be a Thom class and t.n/ its restriction. Let t1 2 h2.CP 1/ and tn 2 h2.CP n/ be the corresponding elements. Since tn is the restriction of t1 let us write just t for all these elements. We have a surjective restriction homomorphism h.CP nC1/! h.CP n/. Thus the restrictions induce an isomorphism (see (17.1.6)and (17.1.7)) h.CP 1/ Š limn h.CP n/ Š limn hŒt=.t nC1/: The algebraic limit is hŒŒt. This shows: (19.1.4) Theorem. h.CP 1/ Š hŒŒt. We extend the previous results by a formal trick to products X CP n. Let p W X CP n! CP n be the projection. We set u D un D p.tn/. (19.1.5) Proposition. Consider h.X CP n/ as a graded h.X/-algebra. Then h.X CP n/ Š h.X/Œu=.unC1/ and h.X CP 1/ Š h.X/ŒŒu. Proof. The cohomology theory k./ D h.X / is additive and multiplicative, and the coefﬁcient algebra is h.X/. The multiplicative structure in k./ is induced by the -product of h./ and the diagonal of X. The element u1 now plays the role of t1. Let pi W.CP 1/n! CP 1 be the projection onto the i-th factor, and set Ti D p i.t1/. Then (19.1.5) implies: (19.1.6) Proposition. h..CP 1/1/ Š hŒŒT1; : : : ; Tn. This statement uses algebraic identities of the type hŒŒx; y Š.hŒŒx/�
.1) Example. Let D nC1 W EnC1! CP n. Then Q./! P./ is canonÞ ically isomorphic to nC1 and t D tn. (19.2.2) Theorem. The h.B/-module h.P.// is free with basis In particular p 1 n1 W h.B/! h.P.// is injective. : Proof. This is a consequence of the Leray–Hirsch theorem (17.8.4) and the com- putation (19.1.3). (19.2.3) Corollary. There exist uniquely determined elements cj./ 2 h2j.B/ such that P n j D0.1/j cj./t nj D 0; since t n is a linear combination of the basis (c0./ D 1). 472 Chapter 19. Characteristic Classes (19.2.4) Remark. Here is a justiﬁcation for the choice of the signs. Let D nC1. Þ Then t tn D 0 and hence c1.nC1/ D tn. (19.2.5) Proposition. Let'W! be a bundle map over f W B! C. Then the naturality relation f.cj.// D cj./ holds. Proof. The homotopy relation k ı P.'/'k implies P.'/t D t. This yields 0 D P.'/ P j.1/j cj./t nj D P j.1/j f.cj.//t nj : Comparing coefﬁcients gives the claim. We have used the rule P.'/.a x/ D f.a/ P.'/.x/, a 2 h.C /, x 2 h.P.// for the module structure; it is a consequence of the naturality of the cup product. (19.2.6) Proposition. Let and be bundles over B. Then the sum formula P cr. ˚ / D iCj Dr ci./cj./: holds. We set ci./ D 0, ifi > dim. Proof. Consider the subspaces P./ P. ˚ / P./ and their open complements U D P. ˚ / X P./ and V D P. ˚
/ X P./. The inclusions P./ U and P./ V are deformation retracts. Let s W P./! P. ˚ / X P./, Œx 7! Œx; 0 and W P. ˚ / X P./! P./, Œx; y 7! Œx. Then s D id and s'id by the homotopy.Œx; y; / 7! Œx; y. Let t D t˚. Consider the elements (k D dim ; l D dim ) P k iD0.1/i ci./t ki ; x D y D P l j D0.1/j cj./t lj : Under the restriction h.P. ˚ //! h.U / Š h.P.// the element x is sent to zero; this is a consequence of the deﬁnition of the ci./, the deformation retraction and the naturality tjP./ D t. Hence x comes from an x0 2 h.P. ˚ /; U /. Similarly y comes from an element y0 2 h.P. ˚ /; V /. Since U; V is an open covering of P. ˚ /, we see that x0y0 D 0 and therefore xy D 0. We use the deﬁnition of the cr. ˚ / in the relation xy D P kCl rD0.1/r P iCj Dr ci./cj./ t r and arrive at the desired sum formula by comparing coefﬁcients. 19.3 Chern Classes Let h./ be a cohomology theory with universal element t D t1 2 h2.CP 1/. Our ﬁrst aim is the computation of h.BU.n//. Recall that BU.1/ D CP 1. The space BU.n/ is the basis of the universal n-dimensional complex vector bundle n 19.3. Chern Classes 473 and 1 D 1. We use that h.BU.1/n/ Š hŒŒT1; : : : ; Tn, see (19.1.6). Let us recall the ring of formal graded power series hŒŒc1; : : : ; cn. The indeterminate c
j has degree 2j. The degree of a monomial in the cj is the sum of the degrees of the factors degree.ck.1/ 1 ck.2/ 2 : : : / D 2k.1/ C 4k.2/ C : A homogeneous power series of degree k is the formal sum of terms of the form j Mj where Mj is a monomial of degree m and j 2 hkm. Thus we assign the degree k to the elements in the coefﬁcient group hk. (19.3.1) Lemma. A classifying map ˇ W BU.n 1/ BU.1/! BU.n/ of the product n1 1 is the projective bundle of n. Proof. Let U.n 1/ U.1/ U.n/ be the subgroup of block diagonal matrices. We obtain a map ˛ W B.U.n 1/ U.1// D EU.n/=.U.n 1/ U.1// D EU.n/ U.n/.U.n/=U.n 1/ U.1//! BU.n/: A model for the universal vector bundle is n W EU.n/ U.n/ Cn! BU.n/. The U.n/-matrix multiplication on Cn induces a U.n/-action on the corresponding projective space P.Cn/. The projective bundle associated to the universal bundle n is EU.n/ U.n/ P.Cn/! BU.n/. We now use the U.n/-isomorphism P.Cn/ Š U.n/=U.n 1/ U.1/. Hence ˛ is the projective bundle of n. We compose with a canonical h-equivalence j W BU.n 1/ BU.1/! B.U.n 1/ U.1//. It remains to show that ˇ D ˛ ı j is a classifying map for n1 1. Let EU.n 1/ EU.1/! EU.n/ be a U.n 1/ U.1/-map. From it we obtain a bundle map E.n1/ E.1/ D.EU.n 1/ EU.1// U.n1/U.1/.Cn1 C1/! EU.n
/ U.n1/U.1/ Cn! EU.n/ U.n/ Cn D E.n/: It is a bundle map over ˛ ı j. (19.3.2) Theorem. Let W BU.1/n! BU.n/ be a classifying map of the n-fold Cartesian product of the universal line bundle. Then the following holds: The induced map W h.BU.n//! h.BU.1/n/ is injective. The image consists of the power series which are symmetric in the variables T1; : : : ; Tn. Let ci 2 h2i.BU.n// be the element such that.ci / is the i-th elementary symmetric polynomial in T1; : : : ; Tn. Then h.BU.n// Š hŒŒc1; : : : ; cn: The elements c1; : : : ; cn are those which were obtained from the projective bundle associated to n by the methods of the previous section. 474 Chapter 19. Characteristic Classes 1 D 1 1. Hence ı is another classifying map of n Proof. Let 2 Sn be a permutation and also the corresponding permutation of the factors of BU.1/n. Then is covered by a bundle automorphism of the n-fold product n 1 and therefore homotopic to. The permutation induces on h.BU.1/n/ D hŒŒT1; : : : ; Tn the corresponding permutation of the Tj. Hence the image of is contained in the symmetric subring, since ı '. Let prj W BU.1/n! BU.1/ be the projection onto the j -th factor. We write.j / D pr 1 D.1/ ˚ ˚.n/ and Tj D c1..j //. We have the relation (naturality) j.1/ so that n ci.n/ D ci.n/ D ci. n 1 / D ci..1/ ˚ ˚.n//: By the sum formula (19.2.6) this equals c1..1//ci1..2/ ˚ ˚.n// C ci..2/ ˚ ˚.n
trivial 1-dimensional bundle; this follows from the sum formula (19.2.6) and ci."/ D 0 for i > 0. This fact suggests that we pass to the limit n! 1. 19.3. Chern Classes 475 (19.3.3) Example. The complex tangent bundle T CP n of the complex manifold CP n satisﬁes T CP n ˚ " Š.n C 1/nC1, see (15.6.6). Therefore the total Chern Þ class of this bundle is.1 C c1.nC1//nC1. Let! W BU.n/! BU.n C 1/ be a classifying map for n ˚ ". Then!ci D ci for i n and!cnC1 D 0. Let U D colimn U.n/, with respect to the inclusions U.n/! U.n C 1/; A 7! ; A 0 1 0 be the stable unitary group. The classifying space BU is called the classifying space for stable complex vector bundles. We think of this space as a homotopy colimit (telescope) over the maps BU.n/! BU.n C 1/. By passage to the limit we obtain (since the lim1-term vanishes by (17.1.7)): (19.3.4) Theorem. h.BU/ Š lim hBU.n/ Š hŒŒc1; c2; : : :. (19.3.5) Example. Let m;n W BU.m/ BU.n/! BU.m C n/ be a classifying map for m n D pr 1 m/ and j D cj.pr c00 2 n. We use the elements c0 2 n/ and obtain j D cj.pr 1 m ˚ pr.1/ h.BU.m/ BU.n// Š hŒŒc0 1; : : : ; c0 m; c00 1 ; : : : ; c00 n: Moreover, by the sum formula,.2/ m;nck D P iCj Dk c0 i c00 j : The map the variables c1; : : : ; cmCn is obtained by inserting for ck the value (2). m;n is continuous in the
��rst Chern class of. Þ (19.3.7) Proposition. The relation c1. ˝/ D c1./Cc1./ holds for line bundles and. Proof. We begin with the universal situation. We know that H 2.CP 1I Z/ Š Z is generated by the ﬁrst Chern class c of the universal bundle D 1. Let k W CP 1 CP 1! CP 1 be the classifying map of O˝. Let prj W CP 1 CP 1! CP 1 be the projection onto the j -th factor. Then H 2.CP 1 CP 1I Z/ has the Z-basis T1; T2 with Tj D pr j.c/. There exists a relation kc1./ D a1e1 C a2e2 with certain ai 2 Z. Let i1 W CP 1! CP 1 CP 1; x 7!.x; x0/ for ﬁxed x0. Then 1 e1 D c1./, since pr1 i1 D id, and i i 1 e2 D 0 holds, since pr2 i1 is constant. We compute a1c1./ D i 1 kc1./ D c1.i 1 k/ D c1.i 1.pr 1 ˝ pr 2 // D c1./; since i 1 pr we see a2 D 1. 1 D and i 1 pr 2 is the trivial bundle. Hence a1 D 1, and similarly We continue with the proof. Let k ; k W B! CP 1 be classifying maps of c1./ and similarly for. With the diagonal d the and. Then c1./ D k equalities ˝ D d. O˝/ D d.k k/. O˝/ hold. This yields c1. ˝ / D c1.d.k k/. O˝// D d.k k/.e1 C e2/ D d.k k/ pr c1./ C k D k D c1./ C c2./; c1./ 1 c1./ C d.k k/ pr 2 c1./ since k D pr1.k k/d holds. 19.3. Chern Classes 477 (19.3.8) Proposition. Let W E./! B be an n-dimensional vector bundle and p W P./! B the associated projective
bundle. The induced bundle splits p./ D 1 ˚ 0 into the canonical line bundle 1 over the projective bundle and another.n 1/-dimensional bundle 0. Proof. Think of as associated bundle E U.n/ Cn! B. Let H be the subgroup U.n 1/ U.1/ of U.n/. We obtain the pullback E H Cn1 C Š E U.n/.U.n/ H Cn/ E U.n/ Cn E=H Š E U.n/.U.n/=H / B and this implies the assertion. We now iterate this process: We consider over P./ the projective bundle P. 0/, et cetera. Finally we arrive at a map f./ W F./! B with the properties: (1) f./ splits into a sum of line bundles. (2) The induced map f./ W h.B/! h.F.// is injective. Assertion (2) is a consequence of (19.2.2). A model for f./ is the ﬂag bundle. The ﬂag space F.V / of the n-dimensional vector space V consists of the sequences (D ﬂags) f0g DV 0 V1 Vn D V of subspaces Vi of dimension i. Let V carry a Hermitian form. Each ﬂag has an orthonormal basis b1; : : : ; bn such that Vi is spanned by b1; : : : ; bi. The basis vectors bi are determined by the ﬂag up to scalars of norm 1. The group U.n/ acts transitively on the set of ﬂags. The isotropy group of the standard ﬂag is the maximal torus T.n/ of diagonal matrices. Hence we can view F.V / as U.n/=T.n/. The ﬂag bundle associated to E U.n/ Cn is then f./ W F./ D E U.n/ U.n/=T.n/ Š E=T.n/! B: We can apply this construction to a ﬁnite number of bundles. (19.3.9) Theorem (Splitting Principle). Let 1; : : : ; k be complex vector bundles over B. Then there exists
a map f W X! B such that f W h.B/! h.X/ is injective and f.j / is for each j a sum of line bundles. We now prove (19.0.1). Consider the exact cohomology sequence of the pair.E.n/; E0.n//. We can use E 0.n/ as a model for BU.n 1/. The projection E.n/! BU.n/ is an h-equivalence. Our computation of h.BU.n// shows that 478 Chapter 19. Characteristic Classes we have a short exact sequence 0 h2n.E.n/; E0.n// h2n.E.n// Š h2n.BU.n// i h2n.E0.n/ Š h2n.BU.n 1//: 0 The element cn lies in the kernel of i. It therefore has a unique pre-image t.n/ 2 h2n.E.n/; E0.n//. For an n-dimensional numerable bundle W E./! B we deﬁne t./ 2 h2n.E./E0.// to be the element t.n/ with a classifying map W! n. Then the elements t./ are natural with respect to bundle maps. From the relation n we conclude (by naturality) t.m/t.n/ D t.m n/ and then t./ t./ D t. / for arbitrary numerable bundles. The element t.1/ 2 h2.E.1/; E0.1// corresponds to the chosen element t1 2 h2.CP 1/. The restriction of t.n/ to 1 1 yields t.1/ t.1/. This is a Thom class, since products of Thom classes are Thom classes. This shows that t.n/ is a Thom class. m;n.cmCn/ D c0 mc00 19.4 Stiefel–Whitney Classes The theory of Chern classes has a parallel theory for real vector bundles. Suppose given an element t1 2 h1.RP 1; / h1.RP 1/ such that its restriction to t1 2 h1.RP 1; / is a generator of this h0-module. Then there exists an isomorphism hŒT =.T nC1/ �
/e.n/ D.1/mne.mCn/ and has the following consequences: 480 Chapter 19. Characteristic Classes (19.5.1) Lemma. Let and be oriented bundles and give the sum orientation. Then t. / D.1/jjjjt./ t./ and e. ˚ / D.1/jjjje./e./. Here jj Ddim. Let W E! B be a real vector bundle. It has a complexiﬁcation C D ˝R C. We take ˚ with complex structure J.x; y/ D.y; x/ on each ﬁbres as a model for C. Let be a complex n-dimensional bundle and R the underlying 2n-dimensional real bundle. If v1; : : : ; vn is a basis of a ﬁbre, then v1; iv1; : : : ; vn; ivn is a basis of the ﬁbre of R, and it deﬁnes the canonical orientation. (19.5.2) Lemma. Let be an oriented n-dimensional real bundle. Then.C/R in our model for C above is isomorphic to.1/n.n1/=2 ˚ as an oriented bundle. The factor indicates the change of orientation, and ˚ carries the sum orientation. (19.5.3) Proposition. Let W E! B be a complex n-dimensional bundle. Consider it as a real bundle with orientation and canonical Thom class induced by the complex structure. Then cn./ D e./ 2 H 2n.BI Z/. Proof. This holds for 1-dimensional bundles by deﬁnition of c1. The general case follows by an application of the splitting principle and the sum formula. We set pi./ D.1/i c2i.C/ 2 H 4i.BI Z/ and call this characteristic class the i -th Pontrjagin class of. The bundle C is isomorphic to the conjugate bundle xC. The relation ci./ D.1/i ci.x/ holds in general for conjugate bundles. Hence the odd Chern classes of C are elements of order 2. This is a reason why we ignore them for the moment. The Pontrjagin classes are by deﬁnition compatible with bundle maps (n
aturality) and they do not change by the addition of a trivial bundle (stability). The next proposition justiﬁes the choice of signs in the deﬁnition of the pj. (19.5.4) Proposition. Let be an oriented 2k-dimensional real bundle. Then pk./ D e./2. Proof. We compute pk./ D.1/kc2k.C/ D.1/ke2k..C/R/ D.1/kC2k.2k1/=2e. ˚ / D e. ˚ / D.1/2k2ke./2 D e./2: We have used (19.5.1), (19.5.2), and (19.5.3). One can remove elements of order 2 if one uses the coefﬁcient ring R D ZŒ 1 2 of rational numbers with 2-power denominator (or, more generally, assumes that 1 2 2 R). The next theorem shows the universal nature of the Pontrjagin classes. 19.5. Pontrjagin Classes 481 (19.5.5) Theorem. Let pj denote the Pontrjagin classes of the universal bundle and e its Euler class. Then H.BSO.2nC1/I R/ Š RŒp1; : : : pn; H.BSO.2n/I R/ Š RŒp1; : : : ; pn1; e: Proof. Induction over n. Let n W ESO.n/ SO.n/ Rn! BSO.n/ be the universal oriented n-bundle and p W BSO.n 1/! BSO.n/ the classifying map of n1 ˚ ". As model for p we take the sphere bundle of n. Then we have a Gysin sequence at our disposal. Write Bn D BSO.n/ for short. Suppose n is even. Then, by induction, H.Bn1/ is generated by the Pontrjagin classes, and p is surjective since the classes are stable. Hence the Gysin sequence decomposes into short exact sequences. Let H n denote the algebra which is claimed to be isomorphic to H.Bn/. And let n W H n! H.Bn
/ be the homomorphism which sends the formal elements pj, e onto the cohomology classes with the same name. We obtain a commutative diagram 0 0 H i.Bn/ n H i n e e H iCn.Bn/ p H iCn.Bn1/ n H iCn n n1 H iCn n1 0 0: By induction, n1 is an isomorphism. By a second induction over i the left arrow is an isomorphism. Now we apply the Five Lemma. In order to start the induction, n! H i.Bn/ is an isomorphism for we note that by the Gysin sequence n W H i i < n. Suppose n D 2m C 1. The Euler class is zero, since we use the coefﬁcient ring R. Hence the Gysin sequence yields a short exact sequence 0! H j.Bn/ p! H j.Bn1/! H j 2m.Bn/! 0: Therefore H.Bn/ is a subring of H.Bn1/ via p. The image of p contains the subring P generated by p1; : : : ; pm. We use pm D e2. The induction hypothesis implies rank H j.Bn1/ D rank P j C rank P j 2m: The Gysin sequence yields rank H j.Bn1/ D rank H j.Bn/ C rank H j 2m.Bn/: The equality rank P j D rank H j.Bn/ is a consequence. If pH j.Bn/ 6D P j then the image would contain elements of the form x C ey 2m. Such an element would be linearly independent of the basis elements of P. This contradicts the equality of ranks. 482 Chapter 19. Characteristic Classes (19.5.6) Example. Let be a complex bundle. Then.R/C is isomorphic to ˚ x. An isomorphism from R ˚ R with the complex structure.x; y/ 7!.y; x/ is given by.x; y/ 7! x C iy p 2 ; ix C y p : 2 P Hence the pi.R/ D.1/i c2i. ˚ x/ D.1/i aCbD2i.1
/bca./cb./. Since SO.2/ D U.1/, an oriented plane bundle has a unique complex structure such Þ that R D. The total Pontrjagin class of is therefore 1 C c1./2. (19.5.7) Example. Let D T CP n denote the complex tangent bundle of CP n. Then R is the real tangent bundle. In order to determine the Pontrjagin classes we use.R/C D ˚ x. The total Chern class of this bundle is.1 C c/nC1.1 c/nC1 D.1c2/nC1 if we write H.CP n/ D ZŒc=.cnC1/ with c D c1.nC1/, see (19.3.3). Hence the total Pontrjagin class of CP n, i.e., of its tangent bundle with the canonical Þ orientation, is.1 C c2/nC1. Problems 1. The Pontrjagin classes are stable. Under the hypothesis of (19.5.5) we obtain in the limit H.BSOI R/ Š RŒp1; p2; : : :. The sum formula pk. ˚ / D iCj Dk pi./pj./ holds (p0 D 1). P 19.6 Hopf Algebras We ﬁx a commutative ring R and work in the category R-MOD of left R-modules. The tensor product of R-modules M and N is denoted by M ˝ N. The natural isomorphism! n ˝ m expresses the commutativity of the tensor product. We have canonical isomorphisms l W R ˝M! M, ˝m 7! m and! m. Co-homology will have coefﬁcients in R, if nothing else is speciﬁed. An algebra.A; m; e/ in R-MOD consists of an R-module A and linear maps m W A ˝ A! A (multiplication), e W R! A (unit) such that m.e ˝ 1/ D l, m.1 ˝ e/ D r. If m.m ˝ 1/ D m.1 ˝ m/ holds, then
the algebra is associative, and if m D m holds, the algebra is commutative. Usually we write m.a ˝ b/ D a b D ab. We use similar deﬁnitions in the category of Z-graded R-modules (with its tensor product and interchange map). (19.6.1) Example. Let X be a topological space. Then the graded R-module H.X/ becomes a (graded) associative and commutative algebra with multiplication m W H.X/ ˝ H.X/! H.X X/! H.X/; 19.6. Hopf Algebras 483 where the ﬁrst map is the -product and the second map is induced by the diagonal d W X! X X. The unit is the map induced by the projection X! P onto a Þ point P. A coalgebra.C; ; "/ in R-MOD consists of an R-module C and linear maps W C! C ˝ C (comultiplication), " W C! R (counit) such that." ˝ 1/ D l 1,.1 ˝ "/ D r 1. If. ˝ 1/ D.1 ˝ / holds, the coalgebra is coassociative, and if D holds, the coalgebra is cocommutative. (19.6.2) Example. Let X be a topological space. Suppose H.X/ is a free R-module. The graded R-module H.X/ becomes a (graded) coassociative and cocommutative coalgebra with comultiplication W H.X/! H.X X/ Š H.X/ ˝ H.X/ where the ﬁrst map is induced by the diagonal d and the isomorphism is the Künneth Þ isomorphism. The counit is induced by X! P. A homomorphism of algebras'W.A; m; e/!.A0; m0; e0/ is a linear map'W A! A0 such that'm D m.' ˝ '/ and e0 D 'e. A homomorphism of coalgebras W.C; ; "/!.C 0; 0; "0/ is a linear map W C!
C 0 such that. ˝ / D 0 and "0 D ". A continuous map f W X! Y induces a homomorphism f W H.Y /! H.X/ of the algebras (19.6.1) and a homomorphism f W H.X/! H.Y / of the coalgebras (19.6.2). The tensor product of algebras.Ai ; mi ; ei / is the algebra.A; m; e/ with A D A1 ˝A2 and m D.m1 ˝m2/.1˝ ˝1/ and e D e1 ˝e2 W R Š R˝R! A1 ˝A2. The multiplication m is determined by.a1 ˝ a2/.b1 ˝ b2/ D a1b1 ˝ a2b2 (with the appropriate signs in the case of graded algebras). The tensor product of coalgebras.Ci ; i ; "i / is the coalgebra.C; ; "/ with C D C1 ˝ C2, comultiplication D.1 ˝ ˝ 1/.1 ˝ 2/ and counit " D "1"2 W C1 ˝ C2! R ˝ R Š R. Let.C; ; "/ be a coalgebra. Let C D Hom.C; R/ denote the dual module. The data m W C ˝ C!.C ˝ C /! C and e W R Š R "! C deﬁne the dual algebra.C ; m; e/ of the coalgebra. (The ﬁrst map is the tautological homomorphism. It is an isomorphism if C is a ﬁnitely generated, projective R-module.) Let.A; m; e/ be an algebra with A a ﬁnitely generated, projective R-module. The data W A m!.A ˝ A/ Š A ˝ A and " W A e! R Š R deﬁne the dual coalgebra.A; ; "/ of the algebra. In the case of graded modules we take the graded dual; if A D.An j
n 2 N0/, then the dual is.An D Hom.An; R/ j n 2 N0/. 484 Chapter 19. Characteristic Classes (19.6.3) Example. Let W H.X/! Hom.H.X/; R/ be the map in the universal coefﬁcient sequence. Then is an isomorphism of the algebra (19.6.1) onto the Þ dual algebra of the coalgebra (19.6.2). (19.6.4) Proposition. Let C be a coalgebra and A an algebra. Then Hom.C; A/ carries the structure of an algebra with product ˛ ˇ D m.˛ ˝ ˇ/, for ˛; ˇ 2 Hom.C; A/, and unit e". The product is called convolution. Proof. The map.˛; ˇ/ 7! ˛ ˝ ˇ is bilinear by construction. The (co-)associativity of m and is used to verify that is associative. The unit and counit axioms yield ˛.e"/ D m.˛ ˝ e"/ D m.1 ˝ e/.˛ ˝ 1/.1 ˝ "/ D ˛: Hence e" is a right unit. A bialgebra.H; m; e; ; "/ is an algebra.H; m; e/ and a coalgebra.H; ; "/ such that and " are homomorphisms of algebras. (Here H ˝ H carries the tensor product structure of algebras.) The equality m D.m ˝ m/.1 ˝ ˝ 1/. ˝ / expresses the fact that is compatible with multiplication. The same equality says that m is compatible with comultiplication. This and a similar interpretation of the identities id D "e, e D.e ˝ e/, m." ˝ "/ D "m is used to show that a bialgebra can, equivalently, be deﬁned by requiring that m and e are homomorphisms of coalgebras. A homomorphism of bialgebras is an R-linear map which is at the same time a homomorphism of the underlying algebras and coalgebras.
An antipode for a bialgebra H is an s 2 Hom.H; H / such that s is a twosided inverse of id.H / 2 Hom.H; H / in the convolution algebra. A bialgebra with antipode is called Hopf algebra. (19.6.5) Example. Let X be an H -space with multiplication W X X! X and neutral element x. Then m W H.X/ ˝ H.X/! H.X X/! H.X/ is an algebra structure on H.X/ with unit induced by fxg X. Suppose H.X/ is a free R-module. Then the algebra structure m and the coalgebra structure (19.6.2) deﬁne on H.X/ the structure of a bialgebra. An inverse for the multiplication induces an antipode. Suppose H.X/ is ﬁnitely generated and free in each dimension. Then W H.X/! H.X X/ Š H.X/ ˝ H.X/ is a coalgebra structure and together with the algebra structure (19.6.1) we obtain a bialgebra. Again an inverse for induces an antipode. The duality isomorphism H.X/! Hom.H.X/; R/ is an isomorphism of the bialgebra onto the dual bialgebra of H.X/. 19.6. Hopf Algebras 485 This situation was studied by Heinz Hopf [91]. The letter for the comultiplication (and even the term “diagonal”) has its origin in this topological context. For Þ background on Hopf algebras see [1], [182], [142], [138]. (19.6.6) Example. The space CP 1 is an H -space with multiplication the classifying map of the tensor product of the universal line bundle. The algebra structure is H.CP 1I Z/ D ZŒc with c the universal Chern class c1. Let ŒCP i 2 H2i.CP 1I Z/ denote the image of the fundamental class of CP i under the homomorphism induced by the embedding CP i! CP 1. The coalgebra structure is determined by.c/ D c ˝ 1 C 1 ˝ c, see (19.3
.7). Since hcn; ŒCP ni D1 (see the proof of (18.7.2)), the dual Hopf algebra H.CP 1I Z/ has an additive basis xi D ŒCP i ; i 2 N0; by dualization of the cohomological coalgebra structure we obtain the multiplicative structure xi xj D.i; j /xiCj with.i; j / D.i C j /Š=.iŠj Š/. Geometrically this means that the map CP i CP j! CP iCj,.Œx0; : : : ; xi ; Œy0; : : : ; yj / 7! Œz0; : : : ; ziCj with zk D aCbDk xayb has degree Þ.i; j /. The comultiplication in H.CP 1/ is.xn/ D iCj Dn xi ˝ xj. P P We generalize the Hom-duality of Hopf algebras and deﬁne pairings. Let A and B be Hopf algebras. A pairing of Hopf algebras is a bilinear map A B! R,.a; b/ 7! ha; b i with the properties: For x; y 2 A and u; v 2 B hxy; ui D hx ˝ y;.u/i; hx; uv i D h.x/; u ˝ v i; h1; ui D".u/; hx; 1i D".x/: The bilinear form h; i on A B induces a bilinear form on A ˝ A B ˝ B by hx ˝ y; u ˝ v i D.1/ jyjjujhx; uihy; v i. This is used in the ﬁrst two axioms. A pairing is called a duality between A; B, if hx; ui D0 for all u 2 B implies x D 0, and hx; ui D0 for all x 2 A implies u D 0. An example of a pairing is the Kronecker pairing H.X/ H.X/! R in the case of an H
-space X. An element x of a bialgebra H is called primitive, if.x/ D x ˝ 1 C 1 ˝ x. Let P.H / H be the R-module of the primitive elements of H. The bracket.x; y/ 7! Œx; y D xy yx deﬁnes the structure of a Lie algebra on P.H /. The inclusion P.H / H yields, by the universal property of the universal enveloping algebra, a homomorphism W U.P.H //! H. For cocommutative Hopf algebras over a ﬁeld of characteristic zero with an additional technical condition, is an isomorphism [1, p. 110]. (19.6.7) Example. A coalgebra structure on the algebra of formal powers series RŒŒx is, by deﬁnition, a (continuous) homomorphism W RŒŒx! RŒŒx1; x2 with. ˝ 1/ D.1 ˝ / and ".x/ D 0. Here RŒŒx1; x2 is interpreted as a completed tensor product RŒŒx1 O˝RŒŒx2. Then is given by the power series.x/ D F.x1; x2/ with the properties F.x; 0/ D 0 D F.0; x/; F.F.x; y/; z/ D F.x; F.y; z//: Such power series F are called formal group laws. Þ 486 Chapter 19. Characteristic Classes Problems 1. The Group algebra. Let G be a group and RG the group algebra. The R-module RG is the free R-module on the set G, and the multiplication RG ˝ RG Š R.G G/! RG is the linear extension of the group multiplication. This algebra becomes a Hopf algebra, if we deﬁne the comultiplication by.g/ D g ˝ g for g 2 G, the counit by ".g/ D 1, and the antipode by s.g/ D g1. Let G be a ﬁnite group and O.G/ the R-algebra of all maps G! R with pointwise addition and
multiplication. Identify O.G G/ with O.G/ ˝ O.G/. Show that the group multiplication m induces a comultiplication D m W O.G/! O.G G/. The data ".f / D f.1/ and s.f /.g/ D f.g1/ complete O.G/ to a Hopf algebra. Evaluation at g 2 G deﬁnes an algebra homomorphism O.G/! R. Show that G is canonically isomorphic to the group A Hom.O.G/; R/ of Problem 2. An element g in a Hopf algebra H is called group-like if.g/ D g ˝ g and ".g/ D 1. The set of group-like elements in H is a group under multiplication. The inverse of g is s.g/. 2. Let D be a Hopf algebra and A a commutative algebra. The convolution product induces on the set A Hom.D; A/ of algebra homomorphisms D! A the structure of a group. 3. Let H be a Hopf algebra with antipode s. Then s is an anti-homomorphism of algebras and coalgebras, i.e., s.xy/ D s.y/s.x/, se D e, "s D s,.s ˝ s/ D s. If H is commutative or cocommutative, then s2 D id. 4. Let H1 and H2 be Hopf algebras and ˛ W H1! H2 a homomorphism of bialgebras. Then ˛ commutes with the antipodes. 5. Let R be a ﬁeld of characteristic p > 0. Let A D RŒx=.xp/. The following data deﬁne a Hopf algebra structure on A W.x/ D x ˝ 1 C 1 ˝ x, ".x/ D 0, s.x/ D x. 19.7 Hopf Algebras and Classifying Spaces P The homology and cohomology of classifying spaces BU, BO, BSO lead to a Hopf algebra which we will study from the algebraic view-point in this section. The polynomial algebra RŒa D RŒa1; a2; : : :
becomes a Hopf algebra with coassociative and cocommutative comultiplication determined by.an/ D pCqDn ap ˝ aq and a0 D 1. We consider the algebra as a graded algebra with ai of degree i. (In the following we disregard the signs which appear in graded situations. Another device would be to assume that the ai have even degree, say degree 2i, or that R has characteristic 2.) Let D.1; : : : ; r / 2 Nr 0 be a multi-index with r components. We use the notation a D a1 r. The monomials of type a (for arbitrary r) form an R-basis of RŒa. The homogeneous component RŒan of degree n is spanned by the monomials a with kk D 1 C 22 C C rr D n.! RŒ˛1; : : : ; ˛n where aj is the j -th elementary symmetric polynomial in the ˛1; : : : ; ˛n. The embedding respects the grading if we give ˛j the degree 1. The image is the subalgebra of symmetric functions. The a with 2 Nn 0 form an R-basis of the symmetric polynomials. We have an embedding RŒa1; : : : ; an 1 : : : ar Another, more obvious, R-basis is obtained by starting with a monomial 19.7. Hopf Algebras and Classifying Spaces 487 ˛I D ˛i1 I J if.j1; : : : ; jn/ is a permutation of.i1; : : : ; in/. The polynomials n and sum over the Sn-orbit of I D.i1; : : : ; in/. Let us write 1 : : : ˛in †I.˛1; : : : ; ˛n/ D P J I ˛J form an R-basis of the symmetric polynomials in RŒ˛1; : : : ; ˛n. The family I D.i1; : : : ; in/ is called a partition of jI j Di 1 C i2 C C in; in the case that I J, we
say that I and J yield the same unordered partition. We can write †I as a polynomial in the a1; : : : ; an and denote it by I.a1; : : : ; an/. The monomials a which are summands of I have degree kk D jI j. Thus I.a1; : : : ; an/ D I.a1; : : : ; an1; 0/ D I.a1; : : : ; an1/ for n >jI j. (19.7.1) Lemma. If I is a partition of k and n k, then I.a1; : : : ; ak/ is independent of n. We consider it as a polynomial in RŒa. In this way we obtain another R-basis of RŒa which consists of the polynomials I. The homogeneous component of degree n is spanned by the I with I an (unordered) partition of n. Consider the formal power series Un D Q n j D1.1 C a1ˇj C a2ˇ2 aB.n/ P j C / 2 RŒaŒŒˇ1; : : : ; ˇn: P j j n. The polynomial B.n/.ˇ1; : : : ; ˇn/ where the sum is taken over the The series has the form multi-indices D.1; : : : ; r / with jj D is symmetric in the ˇ1; : : : ; ˇn. Hence we can write it as polynomial b.n/.b1; : : : ; bn/ where bk is the k-th elementary symmetric polynomial in the variables ˇ1; : : : ; ˇn. For D.1; : : : ; r / let I./ denote the multi-index.i1; : : : ; im/ with i D j for.e., we begin with 1 entries 1, then 2 m entries 2 and so on; hence m D 1 C C r D jj and kD1 ik D jI./j D kk, i.e., I./ is a partition of kk with (weakly) increasing components. In the notation introduced above B.
n/ b.n/.b1; : : : ; bn/ D I./.b1; : : : ; bn/:.ˇ1; : : : ; ˇn/ D †I./.ˇ1; : : : ; ˇn/; P The polynomial b.n/ only involves the variables b1; : : : ; bjI./j and is independent of n for n jI./j. We denote this stable version by b. The b form an R-basis of the symmetric polynomials in RŒˇ. In this sense we can write formally Q P P 1 iD1.1 C a1ˇj C a2ˇ2 j C / D where U Œn is the ﬁnite partial sum over the with kk Dn (although the inﬁnite product itself is not deﬁned). The reader may verify ab D n0 U Œn U Œ1 D a1b1 U Œ2 D a2 U Œ3 D a3 1b2 C a2b2 1b3 C a3b3 1 2a2b2 1 C a1b1a2b2 C 3a3b3 3a1a2b3 3a3b1b2: 488 Chapter 19. Characteristic Classes The polynomials U Œn are symmetric in the a’s and the b’s U Œn.a1; : : : ; anI b1; : : : ; bn/ D U Œn.b1; : : : ; bnI a1; : : : ; an/: In order to see this note that U D limm;n Um;n with Um;n D D Q Q Q m n j D1.1 C ˛i ˇj / iD1 m j D1.1 C a1ˇj C C amˇm j / D P D b (19.7.2) Lemma. Let us write.a/ D Then a. Proof. The deﬁnition of the b compute P P a ˝ a P b.n/ D Q n iD1.1 C b1˛i C C bn˛n
i /: a a ˝a and b b D P b b. implies the relation b.n/ b.n/ D P b b.n/. We ; a Q Q.a/b.n/ i C / D D D D.1 C.a1/ˇi C.a2/ˇ2.1 C.a1 ˝ 1/ˇi C.a2 ˝ 1/ˇ2 Q P.1 C.1 ˝ a1/ˇi C.1 ˝ a2/ˇ2 P.a ˝ 1/b.n/.1 ˝ a /b.n/ P b.n/.a ˝ a /b.n/ b.n/. The sum is Now we compare coefﬁcients and obtain ﬁnite in each degree. We pass to the stable values b and compare again coefﬁcients. D P a b.n/ P b b.n/ Let Hom.RŒa; R/ be the graded dual of RŒa. We can view this as the module of R-linear maps RŒa! R which have non-zero value only at a ﬁnite number of monomials. Let a. The Hopf algebra structure of RŒa induces a Hopf algebra structure on Hom.RŒa; R/. The basic algebraic result of this section is that the dual Hopf algebra is isomorphic to the original Hopf algebra. be the dual of a, i.e., a.a / D ı (19.7.3) Theorem. The homomorphism ˛ W Hom.RŒa; R/! RŒb; f 7! P f.a/b is an isomorphism of Hopf algebras. The generator bj is dual to aj ˛..ai 1// D bi. 1, that is, Proof. The dual basis element of a is mapped to b. Therefore ˛ is an R-linear isomorphism. It remains to show that ˛ is compatible with the multiplication and the comultiplication. 19.7. Hopf Algebras and Classifying Spaces 489 We verify that ˛ is a homomorphism of algebras.
P P P ˛.f /˛.g/ D f.a /b g.a /b D ; f.a /g.a / P b ; b : The coefﬁcient of b in ˛.f g/ is.f g/.a/ and.f g/.a/ D.f ˝g/.a/ D.f ˝g/ P ; a a ˝a P D ; a f.a /g.a /: Now we use the equality (19.7.2). The deﬁnition of the comultiplication in Hom.RŒa; R/ gives for the element which is dual to a the relation a a.a ˝ a / D a.a C / D ( 1; C D ; otherwise: 0; P / D CD a 1// D bi, the genThis means that.a erators of the algebras Hom.RŒa; R/ and RŒb have the same coproduct. Since we know already that ˛ is a homomorphism of algebras, we conclude that ˛ preIn particular we also have for the b the formula serves the comultiplication. CD b ˝ b..b/ D. Since ˛..ai ˝ a P The Hopf algebras which we have discussed have other interesting applications, e.g., to the representation theory of symmetric groups, see [113]. (19.7.4) Remark. If we deﬁne ˛ in (19.7.3) on theR -module of all R-linear maps, then the image is the algebra RŒŒb of formal power series. The homogeneous Þ components of degree n in RŒb and RŒŒb coincide. (19.7.5) Remark. The duality isomorphism (19.7.3) can be converted into a symmetric pairing Q˛ W RŒb˝RŒa! R. The pairing is deﬁned by Q˛.˛' ˝y/ D '.y/ Þ and satisﬁes Q˛.b ˝ a / D ı. Let'W RŒa!
R be a homomorphism of R-algebras. We restrict'to the component of degree n and obtain 'n W RŒan! R. We identify'with the family.'n/. The duality theorem sets up an isomorphism ˛ W Hom.RŒan; R/ Š RŒbn with the homogeneous part RŒbn of RŒb. A graded group-like element K of RŒb is deﬁned to be a sequence of polynomials.Kn.b1; : : : ; bn/ j n 2 N0/ with K0 D 1 and Kn 2 RŒbn of degree n such that.1/ Kn D P iCj Dn Ki ˝ Kj : Since is a homomorphism of algebras, the relation Kn.b1; : : : ; bn/ D Kn.b1; : : : ; bn/ 490 Chapter 19. Characteristic Classes holds. The comultiplication has the form bn D If we use two independent sets.b0 condition (1) in the form i / and.b00 iCj Dn bi ˝ bj (with b0 D 1). i / of formal variables, we can write the P 1; b0 2 C b0 1b00 1 C b00 2; : : : ; P iCj Dn b0 i b00 j / Kn.b0 1 C b00 P D iCj Dn Ki.b0 1; : : : b0 i /Kj.b00 1; : : : ; b00 j :/ The simplest example is Kn D bn. (19.7.6) Proposition. The sequence.'n/ is an R-algebra homomorphism if and only if the sequence.Kn/ with Kn D ˛.'n/ is a graded group-like element. Proof. We use the duality pairing (19.7.5), now with the notation Q˛.x ˝ y/ D hx; y i. Let.Kn/ be group-like and deﬁne a linear map 'n W RŒan! R by 'n.y/ D hKn; y i. Then for x 2 RŒai and y 2 RŒaj with i C j
D n 'n.xy/ D hKn; xy i D hKn; x ˝ y i D hKi ; x ihKj ; y i D' i.x/'j.y/: Hence.'n/ is an algebra homomorphism. Conversely, let'W RŒa! R be an algebra homomorphism with restriction 'n W RŒan! R in degree n. We set Kn D ˛.'n/. A similar computation as above shows that.Kn/ is a group-like element. (19.7.7) Remark. The algebra homomorphisms'W RŒa! R correspond to families of elements.i 2 R j i 2 N/ via'7!.'.ai / D i /. Given a family.i / the corresponding group-like element is obtained as follows. From Q i.1 C 1ti C 2t 2 i C / D P b D P '.a/b D ˛.'/ we see that Kn.b1; : : : ; bn/ is the component of degree n in P b. Þ We now return to classifying spaces and apply the duality theorem (19.7.3). We have the Kronecker pairing W H.BOI F2/˝H.BOI F2/! F2; x ˝y 7! hx; y i and the duality pairing (19.7.5) Q˛ W F2Œw ˝ F2Œu! F2, now with variables w; u in place of a; b. We also have the isomorphism W F2Œw Š H.BOI F2/ from the determination of the Stiefel–Whitney classes. We obtain an isomorphism of Hopf algebras W F2Œu! H.BOI F2/ determined via algebraic duality by the compatibility relation hx; y i D Q˛.x ˝ y/. The generators of a polynomial algebra are not uniquely determined. Our algebraic considerations produce from the universal Stiefel–Whitney classes as canonical generators of H.BOI F2/ canonical generators of H.BOI F2/ via. In a similar manner we obtain isomorphisms H.BUI Z/ Š ZŒd1;
d2; : : : (vari- ables c; d ) and H.BSOI R/ Š RŒq1; q2; : : : (variables p; q). 19.8. Characteristic Numbers 491 Problems 1. Verify the following polynomials b for kk D4 D jI./j: 2 C 4b1b3 4b4; 1b2 C 2b2 b.0;0;0;1/ D b4 b.1;0;1/ D b2 b.0;2/ D b2 b.2;1/ D b1b3 4b4; 1 4b2 1b2 2b2 2 2b1b3 C 2b4; 2 b1b3 C 4b4; b.4/ D b4; I./ D.4/I I./ D.1; 3/I I./ D.2; 2/I I./ D.1; 1; 2/I I./ D.1; 1; 1; 1/: These b are the coefﬁcients of a in U Œ4 D the a’s and b’s. 2. The assignment RŒa ˝ RŒb! R, a ˝ b 7!.a /.b/ D ı (The formal element U D P ab could be called a symmetric copairing.) ab. Check that U Œ4 is symmetric in is a symmetric pairing. P 19.8 Characteristic Numbers Let W X! BO.n/ be a classifying map of an n-dimensional bundle. It induces a ring homomorphism H.BO.n/I F2/! H.XI F2/. We can also pass to the W H.BOI F2/! H.XI F2/. stable classifying map X! BO and obtain This homomorphism codiﬁes the information which is obtainable from the Stiefel– Whitney classes. We use the isomorphism F2Œw Š H.BOI F2/ and the duality theorem (19.7.3). We use a slightly more general form. Let S be a graded Ralgebra; the grading should correspond to the grading of RŒa, there are no signs. We obtain a graded algebra HomR.
RŒa; S / where the component of degree k consists of the homomorphisms of degree k. The product in this algebra is deﬁned by convolution. Then we have: (19.8.1) Theorem. There exists a canonical isomorphism ˛ W HomR.RŒa; S / Š S ŒŒb of graded R-algebras. Here S ŒŒb D S ŒŒb1; b2; : : : is the algebra of graded formal power series in the bi of degree i. The isomorphism ˛ sends the R-homoP morphism'W RŒa! S to the series '.a/b. In our example we obtain from W H.BOI F2/! H.XI F2/ a series v./ 2 H.XI F2/ŒŒu of degree zero. The constant term is 1, the multiplicativity relation v. ˚ / D v./v./ and the naturality v.f / D f v./ hold. For a line bundle we have v./ D 1 C w1./u1 C w1./2u2 C. These properties characterize the assignment 7! v./. We can apply a similar process to oriented or complex bundles. In the case of a complex oriented theory h./ we obtain series v./ 2 h.X/ŒŒd which are 492 Chapter 19. Characteristic Classes natural, multiplicative and assign to a complex line bundle the series v./ D 1 C c1./d1 C c1./2d2 C. Interesting applications arise if we apply the process to the tangent bundle of a manifold. Let us consider oriented closed n-manifolds M with classifying map M W M! BSO of the stable oriented tangent bundle. We evaluate the homomorphism M on the fundamental class ŒM H n.BSOI R/! H n.M I R/! R; x 7! M.x/ŒM : By the Kronecker pairing duality Hn.BSOI R/ Š HomR.H n.BSOI R/; R/ this homomorphism corresponds to an element in Hn.BSOI R/, and this element is M ŒM, the image of the fundamental class ŒM 2
Hn.M I R/ under.M /, by the naturality h M.p/; ŒM i D hp;.M /ŒM i of the pairing. Under the isomorphism W RŒq1; q2; : : : Š H.BSOI R/ the element M ŒM corresponds to an element that we denote SO.M / 2 RŒq1; q2; : : :. From the deﬁnitions we obtain: (19.8.2) Proposition. Let M denote the oriented tangent bundle of M. Then SO.M / D hv.M /; ŒM i; the evaluation of the series v.M / on the fundamental class. If p 2 H n.M / is a polynomial of degree n in the Pontrjagin classes, then the element (number) pŒM is called the corresponding Pontrjagin number. In a similar manner one deﬁnes a Stiefel–Whitney number by evaluating a polynomial in the Stiefel–Whitney classes on the fundamental class. A closed n-manifold M has an associated element O.M / 2 F2Œu1; u2; : : : Š Hn.BOI F2/, again the image of the fundamental class under the map induced by the stable classifying map M W M! BO of the tangent bundle. (19.8.3) Example. Let us consider M D CP 2k. The stable tangent bundle is 2kC1 where is is the canonical complex line bundle, now considered as oriented bundle; see (15.6.6). By the multiplicativity of the v-classes, we have for the tangent bundle 2k of CP 2k the relation v.2k/ D v./2kC1 D.1 C p1./q1 C p1./2q2 C / 2kC1 D.1 C c2q1 C c4q2 C / 2kC1 where as usual H.CP 2kI R/ Š RŒc=.c2kC1/. Note that p1./ D c2, by (19.5.6). The evaluation on the fundamental class yields the coefﬁcient of c2k in this series, since hc2
n. This allows us to determine the coefﬁcients i : The power series H.c/ D 1 C 1c2 C 2c4 C has the property that the coefﬁcient of c2n in H.c/2nC1 is 1. Hirzebruch [81, p. 14] has found this power series i.1 C 1ti C 2t 2 Q Q H.c/ D c tanh c D 2c e2c 1 C c D 1 C B1 2Š.2c/2 B2 4Š.2c/4 C B3 6Š.2c/6 where the Bj are the so-called Bernoulli numbers. The ﬁrst four values are 1 30 ; B4 D ; B2 D ; B3 D B1 D 1 30 1 42 1 6 : The corresponding coefﬁcients in the power series are 1 D 1 3 ; 2 D 1 32 5 ; 3 D 2 33 5 7 ; 4 D 1 33 52 7 : From these data we obtain the polynomials Ln if we insert in the universal polynomials U Œn.p1; : : : ; pnI q1; : : : ; qn/ for qj the value j. We have already listed the polynomials U Œ1, U Œ2, U Œ3, U Œ4. The result is L1 D L2 D L3 D L4 D p1; 1 3 1 45 1 945 1 14175.7p2 p2 1/;.62p3 13p2p1 C 2p3 1/;.381p4 71p3p1 19p2 2 C 22p2p2 1 3p4 1/: 494 Chapter 19. Characteristic Classes The polynomials Ln are called the Hirzebruch L-polynomials. Problems 1. Show that SO.M N / D SO.M /SO.N /. 2. Let M be the oriented boundary of a compact manifold. Then SO.M / D 0. (See the bordism invariance of the degree.) 3. Show that O.RP 2n/ D u2n modulo decomposable elements. Therefore these elements can serve as polynomial generators of H.BOI F2/ Š F2
Œu in even dimensions. 4. The convolution product of the homomorphisms deﬁned at the beginning of the section satisﬁes 5. Determine SO.CP 2/ and SO.CP 4/. D ˚. Chapter 20 Homology and Homotopy We begin this chapter with the theorem of Hurewicz which says in its simplest form that for a simply connected space the ﬁrst non-zero homotopy group is isomorphic to the ﬁrst non-zero integral homology group. In the case of the sphere S n this is essentially the Hopf degree theorem. In our proof we use this theorem and other consequences of the homotopy excision theorem. We indicate an independent proof which only uses methods from homology theory and the Eilenberg subcomplexes introduced earlier. The theorem of Hurewicz has the important consequence that a map between simply connected CW-complexes is a homotopy equivalence if it induces an isomorphism of the integral homology groups (theorem of Whitehead). Another application is to the geometric realization of algebraic chain complexes as cellular chain complexes. We will see that under suitable hypotheses we do not need more cells in a homotopy type than the homology groups predict. Since homotopy groups are difﬁcult to compute it is desirable to have at least some qualitative information about them. One of the striking results is the famous theorem of Serre that the homotopy groups of spheres are ﬁnite groups, except in the few cases already known to Hopf; in particular the stable homotopy groups of spheres are ﬁnite (except n.S n/). Since for a ﬁnite abelian group A the tensor product A ˝ Q D 0 and since homology theories are objects of stable homotopy, this theorem has the remarkable consequence that rationalized homology theories h./ ˝ Q can be reduced to ordinary rational homology. Along the way we obtain qualitative results in general. They concern, for instance, statements about ﬁniteness or ﬁnite generation and are based on qualitative generalizations of the theorem of Hurewicz. For the expert we point out that we do not use the theory of spectral sequences for the proofs. Only elementary methods like induction over skeleta enter. A basic technical theorem relates in a qualitative manner the homology
of the total space, ﬁbre and base of a ﬁbration. On the algebraic side we use so-called Serre classes of abelian groups: Properties like “ﬁnite generation” are formalized. (In the long run this leads to localization of spaces and categories.) 20.1 The Theorem of Hurewicz The theorem of Hurewicz relates the homotopy and the homology groups of a space. In this section H denotes integral singular homology. Let.X; A; / be a pointed pair of spaces. 496 Chapter 20. Homology and Homotopy We deﬁne natural homomorphisms, called Hurewicz homomorphisms, h.X;A;/ D h W n.X; A; /! Hn.X; A/; n 2; n 1; h.X;/ D h W n.X; /! Hn.X/; such that the diagrams n.X; / n.X; A; / h Hn.X/ h Hn.X; A/ @ @ n1.A; / h Hn1.A/ commute (compatibility with exact sequences). For this purpose we use the definition n.X; / D ŒS.n/; X0 and n.X; A; / D Œ.D.n/; S.n 1//;.X; A/0 of the homotopy groups (see (6.1.4)). We choose generators zn 2 Hn.S.n// and Qzn 2 Hn.D.n/; S.n 1// such that @Qzn D zn1 and q.Qzn/ D zn, where q W D.n/! D.n/=S.n 1/ D S.n/ is the quotient map. If we ﬁx z1, then the other generators are determined inductively by these conditions. We deﬁne h W n.X; A; /! Hn.X; A/ by Œf 7! f.Qzn/ and h W n.X; /! Hn.X/ by Œf 7! f.zn/. With our choice of generators the diagram above is then commutative. From (10.4.4) and the analogous result for the
relative homotopy groups we see that the maps h are homomorphisms. The singular simplex 1! I =@I D S.1/,.t0; t1/ 7! t1 represents a generator z1 2 H1.S.1//. If we use this generator, then h W 1.X; /! H1.X/ becomes the homomorphism which was shown in (9.2.1) to induce an isomorphism 1.X; /ab Š H1.X/ for 0-connected X. Recall that we have a right action of the fundamental group n.X; A; / 1.A; /! n.X; A; /;.x; ˛/ 7! x ˛ via transport. We denote by # n.X; A; / the quotient of n.X; A; / by the normal subgroup generated by all elements of the form x x ˛ (additive notation in n). Recall from (6.2.6) that # 2 is abelian. Representative elements in n which differ by transport are freely homotopic, i.e., homotopic disregarding the base point. Therefore the Hurewicz homomorphism induces a homomorphism h# W # n.X; A; /! Hn.X; A/: The transport homomorphism n.X; A; a1/! n.X; A; a2/ along a path from a1 to a2 induces an isomorphism of the # n-groups, and this isomorphism is independent of the choice of the path. We can use this remark: An unpointed map.D.n/; S.n 1//!.X; A/ yields in each of the groups # n.X; A; a/ a well-deﬁned element (A path connected). Thus, if 1.A/ is trivial, we can regard # n.X; A/ as the homotopy set Œ.D.n/; S.n 1//;.X; A/. The group # 1.X; / is deﬁned to be the abelianized group 1.X; /ab, i.e., the quotient by the commutator subgroup. We set # n.X; / D n.X; / for n 2
and again we have the Hurewicz homomorphism h# W # n.X; /! Hn.X/. 20.1. The Theorem of Hurewicz 497 (20.1.1) Theorem (Hurewicz). Let the space X be.n 1/-connected (n 1). Then h# W # n.X; /! Hn.X/ is an isomorphism. Proof. We have already proved the theorem in the case that n D 1. So let n 2 and then # n D n. Since weak homotopy equivalences induce isomorphisms in homotopy and homology, we only need to prove the theorem for CW-complexes X. We can assume that X has a single 0-cell and no i-cells for 1 i n 1, see n.X nC1/ (8.6.2). The inclusion X nC1 X induces isomorphisms # and Hn.X/ Š Hn.X nC1/. Since the Hurewicz homomorphisms h form a natural transformation of functors, it sufﬁces to prove the theorem for.n C 1/-dimensional complexes. In this case X is h-equivalent to the mapping cone of a map of the form'W A D n.X For X D S n the theorem holds by (10.5.1). By naturality and additivity it then S n k. W holds for pointed sums S n j. We have a commutative diagram n.A/ n.B/ n.X/ Hn.A/ Hn.B/ Hn.X/ 0 0 with exact rows. The exactness of the top row is a consequence of the homotopy excision theorem. (20.1.2) Corollary. Let X be simply connected and suppose that i < n. Then i.X; / D 0 for i < n and h W n.X; / Š Hn.X/. QHi.X/ D 0 for Proof. (20.1.1) says, in different wording, that h W j.X/ Š Hj.X/ for the smallest j such that k.X/ D 0 for 1 k < j. (20.1.3) Theorem. Let.X; A/ be a pair of simply connected CW-complexes. Suppose
Hi.X; A/ D 0 for i < n, n 2. Then i.X; A/ D 0 for i < n and h W n.X; A/! Hn.X; A/ is an isomorphism. Proof. Induction over n 2. We use a consequence of the homotopy excision theorem: Let A be simply connected and i.X; A; / D 0 for 0 < i < n. Then n.X; A; /! n.X=A; / is an isomorphism. The theorem of Seifert and van Kampen shows 1.X=A/ D feg. From Hi.X; A/ D QHi.X=A/ and (20.1.2) we conclude i.X=A/ D 0 for i < n. Let n D 2. Since X and A are simply connected, 1.X; A; / D 0 and the diagram 2.X; A; / Š 2.X=A; / h H2.X; A/ Š Š H2.X=A/ shows that h is an isomorphism. 498 Chapter 20. Homology and Homotopy By induction we know that j.X; A; / D 0 for 0 < j < n. Then a similar diagram shows that h W n.X; A; /! Hn.X; A/ is an isomorphism. (20.1.4) Theorem. Let f W X! Y be a map between simply connected spaces. Suppose f W Hi.X/! Hi.Y / is bijective for i < n and surjective for i D n (n 2). Then f W i.X/! i.Y / is bijective for i < n and surjective for i D n. Proof. We pass to the mapping cylinder and assume that f is an inclusion. The hypothesis is then equivalent to Hi.Y; X/ D 0 and the claim equivalent to i.Y; X/ D 0 for i < n. Now we use (20.1.3). Recall: A map f W X! Y between C W -complexes is an h-equivalence if and only if f W i.X/ Š i.Y / for each i. Together with (20.1.4) we obtain a homological version of this result: (
20.1.5) Theorem (Whitehead). Let f W X! Y be a map between simply connected CW-complexes, which induces isomorphisms of homology groups. Then f is a homotopy equivalence. In (20.1.5) one cannot dispense with the hypothesis that the spaces are simply connected. There exist, e.g., so-called acyclic complexes X with reduced homology groups vanishing but with non-trivial fundamental group. Moreover it is important that the isomorphism is induced by a map. (20.1.6) Proposition. Let X be a simply connected CW-complex with integral homology of a sphere, H.X/ Š H.S n/, n 2. Then X is h-equivalent to S n. Proof. By (20.1.2), n.X/ Š Hn.X/, and this group is assumed to be isomorphic to Z. Let f W S n! X represent a generator. Then f W n.S n/! n.X/ is an isomorphism and also f W Hn.S n/! Hn.X/. Now we use (20.1.5). The preceding proposition has interesting applications. It is known that a closed connected n-manifold of the homotopy type of the n-sphere is actually homeomorphic to the n-sphere. Therefore these spheres are characterized by invariants of algebraic topology. (20.1.7) Example. The spaces S n _ S n _ S 2n and S n S n are for n 2 simply connected and have isomorphic homology groups. But they are not h-equivalent, Þ since their cohomology rings are different. The homological theorem of Whitehead (20.1.5) no longer holds for spaces which are not simply connected, even in the case when the map induces an isomorphism of the fundamental groups. But it sufﬁces to consider the universal covering, as the next theorem shows. 20.1. The Theorem of Hurewicz 499 (20.1.8) Theorem. Let f W X! Y be a map between connected CW-complexes QX! X and which induces an isomorphism f W 1.X/! 1.Y /. Let p W q W QY! Y be the universal coverings. There
exists a lifting F W QX! QY of f, i.e., qF D fp. Suppose F induces an isomorphism of the homology groups. Then f is a homotopy equivalence. Proof. We choose isomorphisms 1.X/ Š G Š 1.Y / which transform f into the identity of G. We then consider p and q as G-principal bundles with left action QX! QY as a G-map. We obtain a morphism of the associated ﬁbre bundles. and F W QX F QY EG G QX EGG F EG G QY BG D BG From the assumption and (20.1.5) we see that F is an h-equivalence. The exact homotopy sequence and the Five Lemma show that EGG F induces isomorphisms of the homotopy groups. We now consider the second associated ﬁbre bundles P W EG G QX! X and Q W EG G QY! Y. A section s of P arises from a map W QX! EG such that.gx/ D.x/g1 for x 2 X and g 2 G, see (14.1.4). A map of this type is essentially the same thing as a classifying map of p. Since the ﬁbre of P is contractible, P induces isomorphisms of the homotopy groups, and the same holds then for a section s of P. We see that f D Qı.EGG F /ıs induces isomorphisms of homotopy groups; hence f is a homotopy equivalence. (20.1.9) Corollary. In the situation of (20.1.8) F is a G-homotopy equivalence. Proof. Let h W Y! X be h-inverse to f and H W QY! QX a lifting of H which is a G-map. A homotopy of hf can be lifted to a G-homotopy of HF. The end of this homotopy is a bundle automorphism. Let G be a discrete group which acts on the pair.Y; B/. The induced maps of the left translations by group elements yield a left action of G on Hn.Y; B/ via homomorphisms, i.e., Hn.Y; B/ becomes a module over the integral
group ring ZG of G. Suppose X is obtained from A by attaching n-cells.n 3/. Let p W Y! X be a universal covering and B D p1.A/. Then Y is obtained from B by attaching n-cells. The group D 1.X/ of deck transformations acts freely on the set of n-cells in Y X B. Hence Hn.Y; B/ is a free Z-module, the basis elements correspond bijectively to the n-cells of X X A. Theorem (20.1.3) now yields: (20.1.10) Theorem. Let X be a connected CW-complex and let n 3. Then n.X n; X n1/ is a freeZ 1.X n1/-module. A basis of this module consists of the characteristic maps of the n-cells. The map h# W n.X n; X n1/! Hn.X n; X n1/ is an isomorphism. 500 Chapter 20. Homology and Homotopy The exact sequence nC1.X nC1; X n/! n.X n; X n1/! n.X nC1; X n1/! 0 is for n 3 a sequence of Z1.X n1/ Š Z1.X n/-modules. Because of the isomorphism n.X; X n1/ Š n.X nC1; X n1/ the sequence is a presentation of the Z1.X n1/-module n.X; X n1/. The induced sequence of the #-groups is also exact. This gives the following theorem for n 3. (20.1.11) Theorem (Hurewicz). Let.X; A/ be a CW-pair with connected X and A. Let.X; A/ be.n 1/-connected (n 2). Then h# W # n.X; A; / Š Hn.X; A/. k Proof. We now give a purely homological proof of the Hurewicz theorems which also covers the relative case n D 2 for spaces which are not simply connected. The proof is by induction. The induction starts with (9.2.1). We assume the absolute theorem for 1 i n 1 and prove the relative theorem for n. We consider the standard simplex Œk D Œe0
; : : : ; ek as the usual simplicial complex and denote its l-skeleton by Œkl. Let S n1.X; A; / be the chain group spanned by simplices W Œk! X such that.Œkn1/ A and.Œk0/ D fg, modulo Sk.A/. Let H.n1/.X; A; / be the n-th homology group of the resulting chain complex. n The inclusion of chain complexes induces an isomorphism H.n1/.X; A; / Š Hn.X; A/ for an.n 1/-connected pair.X; A/ with path connected A, see (9.5.4). We have to adapt the homotopy groups to the simplicial setup. We consider elements of n.X; A; / as homotopy classes of maps f W.Œn; @Œn; e0/!.X; A; /. For this purpose we ﬁx a homeomorphism ˛ W.D.n/; S.n 1/; /!.Œn; @Œn; e0/ which sends the generator Qzn deﬁned at the beginning to the standard generator Œid.Œn/ 2 Hn.Œn; @Œn/. n The Hurewicz homomorphism then sends the homotopy class of f to fŒid,.X; A; /. We now construct an inverse and this class is an element in H.n1/ W H.n1/.X; A; /! # n n n.X; A; / of h. We assign to a singular simplex W.Œn; @Œn; Œn0/!.X; A; / n.X; A; / represented by. the element in # If.Œn/ A, then the corresponding homotopy class is zero. Since the # n-group is abelian, we obtain a well-deﬁned homomorphism W S n1 n.X; A; /. The simplices are cycles (since S n1 n1.X; A; / D 0), and thus it remains to show that the composite ı @ W S n1 n.X; A
; / is trivial, in order to obtain. We reduce the problem to a universal situation. For this purpose we deﬁne elements bn 2 n.@Œn C 1; Œn C 1n1; e0/, 0 Œe1e0/Œd 3 nC1.X; A; /! #.X; A; /! # b2 D.Œd 3 1 1Œd 3 3 1; n 2 Œd 3 nC1X bn D Œd nC1 0 Œe1e0 C iD1.1/i Œd nC1 i ; n 3; 20.2. Realization of Chain Complexes 501 where Œvw denotes the afﬁne path class in Œn C 1 from v to w, and we use the transport along this path. (Multiplicative notation in 2, additive notation for n 3. This deﬁnition corresponds to the homological boundary operator, but we have to transport the face maps to the base point e0. For b2 we have to pay attention to the order of the factors, since the group is non-abelian.) Let us write K D Œn C 1 and let W.K; Kn1; K0/!.X; A; / be a basis element of S n1 nC1.X; A; /. Then P @Œ D i.1/i Œd nC1 i D # Œbn# D # j # Œbn# where j denotes the inclusion @Œn C 1 Œn C 1. Thus it remains to show that jŒbn D 0. The skeleton Kn1 is.n 2/-connected (use e.g., the induction hypothen1.Kn1; e0/! Hn1.Kn1/ is sis). Hence, by the inductive assumption, # an isomorphism. The commutativity @h D h@ now shows that @Œbn D 0, since @hŒbn D 0 by the fundamental boundary relation for singular homology. We have the factorization @ W n.Kn; Kn1; e0/ j! n.K; Kn1; e0/ @0! n1.Kn1; e0/; and @0 is an isomorphism, since K
is contractible. This ﬁnishes the inductive step for the relative Hurewicz theorem. For n 2 the absolute theorem is a special case of the relative theorem. An interesting consequence of the homological proof of the Hurewicz theorem is a new proof of the Brouwer–Hopf degree theorem n.S n/ Š Z. 20.2 Realization of Chain Complexes The computation of homology groups from the cellular chain complex shows that one needs enough cells to realize the homology groups algebraically as the homology groups of a chain complex. It is interesting to know that in certain cases a converse holds. We work with integral homology. (20.2.1) Theorem (Cell Theorem). Let Y be a 1-connected CW-complex. Suppose Hj.Y / is ﬁnitely generated for j n. Then Y is homotopy-equivalent to a CWcomplex Z with ﬁnitely many j -cells for j n. The proof of this theorem is based on a theorem which says that under suitable hypotheses an algebraic chain complex can be realized as a cellular chain complex. We describe the inductive construction of a realization. We start with the following: 20.2.2 Data and notation. (1) Y is a CW-complex with i-skeleton Yi. 502 Chapter 20. Homology and Homotopy (2) Zr is an r-dimensional CW-complex. (3) f W Zr! Y is a cellular map. (4) Ci.Z/ D Hi.Zi ; Zi1I Z/ is the i-th cellular chain group. (5) f induces a chain map'W C.Z/! C.Y /. (6) We attach.r C 1/-cells to Zr such that f can be extended to F : qS r j f Zr \ Yr \ qDrC1 j ZrC1 F YrC1. (7) From this diagram we obtain a resulting diagram of chain groups ArC1 ı CrC1.Y / d 0 rC1 Cr.Z/ 'r Cr.Y / d d 0 r Cr1.Z/ 'r1 Cr1.Y / with ArC1 D HrC1.ZrC1; Zr / a free abelian group with a basis given by the Þ.r C 1/-cells
, and induced by.F; f /. We now start from a diagram in which ArC1 is a free abelian group with basis.aj j j 2 J /. The horizontal parts should be chain complexes, i.e., d ı D 0. Can this diagram be realized geometrically? (20.2.3) Proposition. A realization exists, if the following holds: (1) f W Hi.Zr /! Hi.Y / is bijective for i r 1 and surjective for i D r; (2) r 2; (3) Zr and Y are 1-connected. Proof. Suppose we are given for each j 2 J a diagram S r DrC1 bj Bj Zr f YrC1. We attach.r C 1/-cells to Zr with attaching maps bj to obtain ZrC1 and use the Bj to extend f to frC1 W ZrC1! YrC1. Then ArC1 Š HrC1.ZrC1; Zr / canonically and basis preserving. We consider f as an inclusion. The assumption (1) is then equivalent to Hi.Y; Zr / D 0 for i r. Since r 2 and 1.Y / D 0 we also have 1.YrC1/ D 0. 20.2. Realization of Chain Complexes 503 Therefore we have the relative Hurewicz isomorphism h W rC1.YrC1; Zr / Š HrC1.YrC1; Zr /, since Hi.YrC1; Zr / Š Hi.Y; Zr / D 0 for i r. The diagram represents an element in rC1.YrC1; Zr /. Let xj D h.ŒBj ; bj / 2 HrC1.YrC1; Zr / be its image under the relative Hurewicz homomorphism. This element can be determined by homological conditions. The correct maps ı and are obtained, if xj has the following properties: (1) The image of xj under @ W HrC1.YrC1; Zr /! Hr.Zr /! Hr.Zr ; Zr1/ is ı.aj /. (2) The image of xj under W H
rC1.YrC1; Zr /! HrC1.YrC1; Yr / is.aj /. We show that there exists a unique element xj with these properties. We know HrC1.Y; r; Zr / D 0 and Hj.Yr1Zr1/ D 0, j r for reasons of dimension. The exact sequence of the triple.YrC1; Yr ; Zr / shows that is injective. Hence there exists at most one xj with the desired properties. The existence follows if we show that Im.@; / D Ker.'r d 0 rC1/. This follows by diagram chasing in the next diagram with exact rows HrC1.YrC1; Zr1/ ˛ HrC1.YrC1; Yr1/ HrC1.YrC1; Zr / @ HrC1.YrC1; Yr / d 0 rC1 'r Hr.Yr ; Yr1/ Hr.Zr ; Zr1/ Hr.YrC1; Zr1/ ˇ Hr.YrC1; Yr1/: One uses that ˛ is surjective and ˇ injective. Proof. Since Y is simply connected, we can assume that Y has a single 0-cell and no 1-cells. We construct Z inductively with Z0 D fg and Z1 D fg. We choose a ﬁnite number of generators for 2.Y / Š H2.Y / and representing maps S 2! Y2. S 2! Y, and the induced map is surjective They yield a cellular map f2 W Z2 D in H2 and bijective in Hj, j < 2. This starts the inductive construction. W r /= Im.d 0 Suppose fr W Zr! Y is given such that fr W Hi.Zr /! Hi.Y / is bijective for i r 1 and surjective for i D r. We construct a diagram of type (7) in 20.2.2 as follows. We have Hr.Zr / D Ker.d / and Hr.Y / D Ker.d 0 rC1/ and the map.fr / W Hr.Zr /
D Ker.d /! Hr.Y / is surjective. Let ArC1 be the kernel of.fr / and ı W Hr.Zr / Cr.Zr /. As a subgroup of Cr.Zr / it is free abelian. Since Zr has, by induction, a ﬁnite number of r-cells, the group ArC1 is ﬁnitely generated. By deﬁnition of ArC1, the image of 'r ı is contained in the image of d 0 rC1. Hence there exists making the diagram commutative. We now apply (20.2.3) in order to attach an.r C 1/-cell for each basis element of ArC1 and to extend fr to f 0 rC1! Y. By construction,.f 0 rC1/ is now bijective on Hr. If this map is not yet surjective on HrC1 we can achieve this by attaching more.r C 1/-cells with trivial attaching maps; if HrC1.Y / is ﬁnitely generated, we only need a ﬁnite number of cells for this purpose. We continue in this manner as long as H.Y / is ﬁnitely generated. After that point we do not care about ﬁnite generation. The ﬁnal map f W Z! Y is a homotopy equivalence by (20.1.5). rC1 W Z0 504 Chapter 20. Homology and Homotopy 20.3 Serre Classes Typical qualitative results in algebraic topology are statements of the type that the homotopy or homology groups of a space are (in a certain range) ﬁnite or ﬁnitely generated or that induced maps have ﬁnite or ﬁnitely generated kernel and cokernel. A famous result of Serre [170] says that the homotopy groups of spheres are ﬁnite, except in the cases already known to Hopf. Here are three basic ideas of Serre’s approach: (1) Properties like ‘ﬁnite’ or ‘ﬁnitely generated’ or ‘rational isomorphism’ have a formal structure. Only this structure matters – and it is axiomatized in the
notion of a Serre class of abelian groups or modules. (2) One has to relate homotopy groups and homology groups, since qualitative results about homology groups are more accessible. The connection is based on the Hurewicz homomorphism. (3) For inductive proofs one has to relate the homology groups of the basis, ﬁbre, and total space of a (Serre-)ﬁbration. This is the point where Serre uses the method of spectral sequences. L A non-empty class C of modules over a commutative ring R is a Serre class if the following holds: Let 0! A! B! C! 0 be an exact sequence of R-modules. Then B 2 C if and only if A; C 2 C. We call C saturated if A 2 C implies that j A of copies of A are contained in C. The class consisting arbitrary direct sums of the trivial module alone is saturated. A morphism f W M! N between Rmodules is a C-epimorphism (C-monomorphism) if the cokernel (kernel) of f is in C, and a C-isomorphism if it is a C-epi- and -monomorphism. We use certain facts about these notions, especially the C-Five Lemma. The idea is to neglect modules in C, or, as one says, to work modulo C; so, instead of C-isomorphism, we say isomorphism modulo C. Here are some examples of Serre classes. (1) The class containing only the trivial group. (2) The class F of ﬁnite abelian groups. (3) The class G of ﬁnitely generated abelian groups. (4) Let R be a principal ideal domain. The class C consists of the (ﬁnitely generated) R-modules. If R is a ﬁeld, then we are considering the class of (ﬁnite-dimensional) vector spaces. (5) Let R be a principal ideal domain. The class C consists of the (ﬁnitely generated) R-torsion modules. A module M is a torsion module, if for each x 2 M there exists 0 ¤ 2 R such that x D 0. (6) Let P N be a set of prime numbers. Let ZP Q
denote the subring of rational numbers with denominators not divisible by an element of P. If P D ;, then Z D Q. If P D fpg, then ZP D Z.p/ is the localization of Z at p. If P contains the primes except p, then ZP D ZŒp1 is the 20.4. Qualitative Homology of Fibrations 505 ring of rational numbers with denominators only p-powers. The rings ZP are principal ideal domains. Let P 0 be the complementary set of primes. An abelian group A is a P 0-torsion group if and only if A ˝Z ZP D 0. Let CP be the class of P 0-torsion groups. Then a homomorphism'W A! B is a CP -isomorphism if and only if'˝Z ZP is an ordinary isomorphism. Similarly for epi- and monomorphism. This remark reduces the CP Five Lemma to the ordinary Five Lemma after tensoring with ZP. This simpliﬁes working with this class1. (20.3.1) Proposition (Five Lemma mod C). In the next proposition we use the same notation as in (11.2.7). The considerations of that section then yield directly a proof of the following assertions. (1) b C-epimorphism ) Qb C-epimorphism. (2) d C-epimorphism, e C-monomorphism ) Qd C-epimorphism. (3) The hypotheses of.1/ and.2/ imply: c C-epimorphism. (4) d C-monomorphism ) Qd C-monomorphism. (5) a C-epimorphism, b C-monomorphism ) Qb C-monomorphism. (6) The hypotheses of.4/ and.5/ imply: c C-monomorphism. The kernel-cokernel-sequence shows other properties of C-notions. (20.3.2) Proposition. Given homomorphisms f W A! B and g W B! C between R-modules. (1) If f and g are C-monomorphisms (-epimorphisms), then gf is a C-mono- morphism (-epimorphism). (2) If two of
the morphisms f, g, and gf are C-isomorphisms so is the third. 20.4 Qualitative Homology of Fibrations In this section we work with singular homology with coefﬁcients in the R-module M. (20.4.1) Theorem (Fibration Theorem). Let p W E! B be a (Serre-) ﬁbration with 0-connected ﬁbres. Let.B; A/ be a relative CW-complex with t-skeleton B t. We assume that s1 for an s 0 and that there are only a ﬁnite number of t-cells for t e. Finally we assume that Hi.F I M / 2 C for 0 < i < r and all ﬁbres F of p. We write Et D p1.B t /. Then the following holds: (1) Let C be saturated. Then p W Hi.E; E1I M /! Hi.B; B 1I M / is a Cisomorphism for i r C s 1 D ˛ and a C-epimorphism for i D ˛ C 1. 1In a more abstract setting one can construct localizations of categories so that C-isomorphisms become isomorphisms in the localized category, etc. 506 Chapter 20. Homology and Homotopy (2) Let C be arbitrary. Then p is a C-isomorphism for i ˛ and a C-epi- morphism for i D ˛ C 1 where now ˛ D min.e C 1; r C s 1/. We remark that r 1. For r D 1 we make no further assumptions about F. Since a weak homotopy equivalence induces isomorphisms in singular homology, we can assume without essential restriction that B is a CW-complex (pull back the ﬁbration along a CW-approximation). We reduce the proof of the theorem by a Five Lemma argument to the attaching of t-cells. We consider the following situation. Let.ˆ; '/ W qa.Dt a; S t1 a be an attaching of t-cells. Let p W E! B be a ﬁbration and set E 0 D p1.B 0/. We assume that the ﬁbres are 0-connected and homot
opy equivalent. We pull back the ﬁbration along ˆ and obtain two pullback diagrams. /!.B; B 0/ qEa ‰ E qpa p qDt a ˆ B qE0 a qp0 a ‰ E0 p0 qS t1 a ˆ B 0 We apply homology (always with coefﬁcients in M ) and obtain the diagram Hi.E; E0/ p Hi.B; B 0/ L ‰ a Hi.Ea; E0 a/ ˚apa L ˆ a Hi.Dt a; S t1 a /: We already know that ˆ is an isomorphism. In order to show that ‰ is an isomorphism, we attach a single t-cell, to simplify the notation. Let B0 be obtained from B by deleting the center ˆ.0/ of the cell. (20.4.2) Lemma. Let p W X! B be a ﬁbration with restrictions p0 W X 0! B 0 and p0 W X0! B0. Let q W Y! Dt be the pullback of p, and similarly q0 W Y 0! S t1 and q0 W Y0! Dt X 0. Then ‰ W hi.Y; Y 0/! hi.X; X 0/ is an isomorphism for each homology theory. Proof. We have a commutative diagram hi.Y; Y 0/.1/ hi.Y; Y0/.3/ hi.Y X Y 0; Y0 X Y 0/ ‰ ‰ ‰.5/ hi.X; X 0/.2/ hi.X; X0/.4/ hi.X X X 0; X0 X X 0/: 20.4. Qualitative Homology of Fibrations 507 Since B 0 is a deformation retract and p a ﬁbration, also X 0 is a deformation retract of X0. Therefore (2) is an isomorphism by homotopy invariance. The map (4) is an isomorphism by excision. For similar reasons (1) and (3) are isomorphisms. Finally (5) is induced by a homeomorphism. By the homotopy theorem for ﬁbrations, a
ﬁbration over Dt is ﬁbre-homotopy equivalent to a product projection Dt F! Dt. We use such equivalences and a suspension isomorphism Hi.Dt F; S t1 F / Š Hit.F / and obtain altogether a commutative diagram (Pa a point and Fa the ﬁbre over Pa): Hi.E; E0/ p Hi.B; B 0/ L Š a Hit.Fa/ L Š a Hit.Pa/. (20.4.3) Note. The considerations so far show that the bottom map has the following properties: (1) Isomorphism for i t (since ﬁbres are 0-connected). (2) Epimorphism always. (3) Suppose Hi.Fa/ 2 C for 0 < i < r. Then each particular map Hit.Fa/! Hit.Pa/ is a C-isomorphism for 0 < i t < r. Thus the total map is a C-isomorphism if either C is saturated or if we attach a ﬁnite number of Þ cells. We apply these considerations to a ﬁbration p W E! B over a relative CWcomplex.B; A/ as in the statement of the theorem. In this situation the previous considerations yield: Let C be saturated. Then p W Hi.Et ; Et1/! Hi.B t ; B t1/ is a C-isomorphism for each t 0, if i < r C s. We only have to consider t s. By (20.4.3), we have a C-isomorphism in the cases i t and i > t > i r. These conditions hold for each t, if s > i r. If, in addition, there are only ﬁnitely many t-cells for t e, then we have a C-isomorphism for arbitrary C and each t, if i min.e C 1; r C s 1/, by the same argument. We ﬁnish the proof of theorem (20.4.1) with: (20.4.4) Lemma. Let.B; A/ be a relative CW-complex with t-skeleton B t. Let p W E! B be a ﬁbration and Et D p1.B t /. Suppose p W Hi.Et
; Et1I M /! Hi.B t ; B t1I M / is a C-isomorphism for each t 0 and each 0 i ˛, then p W Hi.E; E1I M /! Hi.B; B 1I M / is a C-isomorphism for 0 i ˛ and a C-epimorphism for i D ˛ C 1. Proof. We show by induction on k 0 that p W Hi.Ek; E1/! Hi.B k; B 1/ is a C-isomorphism (k 0). For the induction step one uses the exact homology 508 Chapter 20. Homology and Homotopy sequence for the triple.Ek; Ek1; E1/, and similarly for B, and applies the CFive Lemma to the resulting diagram induced by the various morphisms p. For the epimorphism statement we also use part (2) in (20.4.3). The colimit k! 1 causes no problem for singular homology. The statement of (20.4.1) is adapted to the method of proof. The hypotheses can be weakened as follows. (20.4.5) Remark. Let.X; A/ be an.s 1/-connected pair of spaces. Then there exists a weak relative homotopy equivalence.B; A/!.X; A/ from a relative CWcomplex.B; A/ with A D B s1. We pull back a ﬁbration over q W E! X along this equivalence and use the fact that weak equivalences induce isomorphisms in singular homology. Then part (1) of (20.4.1) yields that q W Hj.E; q1.A/I M /! Hj.X; AI M / is a C- isomorphism for j r C s 1 and a C-epimorphism for Þ i D r C s. (20.4.6) Remark. Let X be a 1-connected space such that Hj.XI Z/ is ﬁnitely generated for i e. Then there exists a weak equivalence B! X such that B has only a ﬁnite number of t-cells for t e. We use this result in the next section. Þ Problems 1. Suppose that Hj.F I
M / D 0 for 0 < j < r. Let B be.s 1/-connected. Then p W Hj.E; F I M /! Hj.B; I M / is an isomorphism for j r C s 1. We can now insert this isomorphism into the exact homology sequence of the pair.E; F / and obtain an exact sequence HrCs1.F I M /! HrC11.EI M /! HrCs1.BI M /!! H1.F I M /! H1.EI M /! H1.BI M /! 0 which is analogous to the exact sequence of homotopy groups. Compare these sequences via the Hurewicz homomorphism.M D Z/. 20.5 Consequences of the Fibration Theorem We use exact sequences and the ﬁbration theorem to derive a number of results. We consider a ﬁbration p W E! B and assume that B and F D p1./ are 0-connected; then E is 0-connected too. We use the notation Z 2 C.r; M /, Hj.ZI M / 2 C for 0 < j < r 1. (M an R-module. In the case that M D Z we write C.r/. For r D 1 there is no condition.) Let F, G denote the class of ﬁnite, ﬁnitely generated abelian groups, respectively. We use homology with coefﬁcients in the R-module M if nothing else is speciﬁed. 20.5. Consequences of the Fibration Theorem 509 (20.5.1) Remark. The homomorphism p W Hj.E; F I M /! Hj.B; I M / is always an isomorphism for j 1 and an epimorphism for j D 2 (see (20.4.1)). The homomorphism p W Hj.EI M /! Hj.BI M / is an isomorphism for j D 0 and an epimorphism for j D 1. (20.5.2) Theorem. Suppose F 2 C.r; M /. Let C be saturated. Suppose B is s-connected. Then p W Hi.E; F /! Hi.B; / is a C
-isomorphism for i r C s and a C-epimorphism for i D r C s C 1. Moreover p W Hi.E/! Hi.B/ is a C-isomorphism for i < r and a C-epimorphism for i D r. Proof. The ﬁrst statement is (20.4.1). For the second statement we use in addition the exact homology sequence of the pair.E; F /. (20.5.3) Theorem. Let C be saturated. (1) F; B 2 C.r; M / ) E 2 C.r; M /. (2) F 2 C.r; M /; E 2 C.r C 1; M / ) B 2 C.r C 1; M /. (3) B 2 C.r C 1; M /; E 2 C.r; M /, B 1-connected ) F 2 C.r; M /. Proof. (1) and (2) are consequences of the ﬁbration theorem and the exact homology sequence of the pair.E; F /. (3) is proved by induction on r. For r D 1 there is nothing to prove. For the induction step consider Hr.B; I M /.1/ Hr.E; F I M / Hr1.F I M / Hr1.EI M / (1) is a C isomorphism for r 1 C s 1 D r, since s D 2 and F 2 C.r 1; M / by induction. From the hypotheses Hr.BI IM /; H r1.EI M / 2 C we conclude Hr1.F I M / 2 C. (20.5.4) Theorem. Let C be arbitrary and assume that B is 1-connected. (1) F; B 2 C.r; M /; B 2 G.r 1/ ) E 2 C.r; M /. (2) F 2 C.r; M /; E 2 C.r C 1; M /; B 2 G.r/ ) B 2 C.r C 1; M /. (3) B 2 C.r C 1; M /; E 2 C.r; M /; B 2 G.r/ ) F 2 C.r; M /. Proof. As for
(20.5.3). The 1-connectedness of B is needed in order to apply the cell theorem (see (20.4.6)). (20.5.5) Corollary. Suppose E is contractible (path ﬁbration over B). Then F'B, the loop space of B. Let B be simply connected. Let C be saturated. Then B 2 C.r C 1; M / if and only if B 2 C.r; M /. Moreover B 2 G.r C 1/ if and only if B 2 G.r/. Similarly for F instead of G. (20.5.6) Proposition. Let A be a ﬁnitely generated abelian group. Then the Eilenberg–Mac Lane spaces K.A; n/ are contained in G.1/. If A is ﬁnite, then K.A; n/ 2 F.1/. Moreover K.A; 1/ 2 G.1/; K.A; 1/ 2 C.1; M / implies K.A; n/ 2 C.1; M /. If C is saturated, then K.A; 1/ 2 C.1; M / implies K.A; n/ 2 C.1; M /. 510 Chapter 20. Homology and Homotopy Proof. We use the path ﬁbration K.A; n 1/! P! K.A; n/ with contractible P and induction with (20.5.3) and (20.5.4). In the case that n D 1, standard constructions yield models for K.A; 1/ with a ﬁnite number of cells in each dimension. One uses K.A1; 1/ K.A2; 1/ D K.A1 A2; 1/, K.Z; 1/ D S 1, and K.Z=m; 1/ D S 1=.Z=m/. Let X be a.k 1/-connected space (k 2). We attach cells of dimension j k C 2 to X in order to kill the homotopy groups j.X/ for j k C 1. The resulting space in an Eilenberg–Mac Lane space K.; k/, D k.X/, and the inclusion W X! K.; k/ induces an isomorphism k./. We pull back the path ﬁb
ration over K.; k/ and obtain a ﬁbration K.; k 1/! Y! X with k-connected Y, and q W Y! X induces isomorphisms j.q/, j > k. This follows from the exact homotopy sequence. If 2 C and C is saturated, then q W Hj.Y I M /! Hj.XI M / is a C-isomorphism for j > 0, by the ﬁbration theorem, since K.; k 1/ 2 C.1/. Similarly for arbitrary C when X is of ﬁnite type. 20.6 Hurewicz and Whitehead Theorems modulo Serre classes Let C be a Serre class of abelian groups with the additional property: The groups Hk.K.A; 1// 2 C whenever A 2 C and k > 0. In this section we work with integral singular homology. (20.6.1) Theorem (Hurewicz Theorem mod C). Suppose X is 1-connected and n 2. Assume that either C is saturated or Hi.X/ is ﬁnitely generated for i < n and C is arbitrary. Then the following assertions are equivalent: (1) ….n/ W i.X; / 2 C for 1 < i < n. (2) H.n/ W Hi.X/ 2 C for 1 < i < n. If ….n/ or H.n/ holds, the Hurewicz homomorphism hn W n.X; /! Hn.X/ is a C-isomorphism. Proof. The proof is by induction on n. (1) Let X! PX f! X be the path ﬁbration with contractible PX. It provides us with a commutative diagram n.X; / h Hn.X; / n.f / n.PX; X; / Hn.f / h h Hn.PX; X/ @ @ n1.X; / h Hn1.X/. 20.6. Hurewicz and Whitehead Theorems modulo Serre classes 511 The boundary maps @ are isomorphisms, since PX is contractible. By a basic property of ﬁbrations n.f / is an isomorphism. By the ordinary Hurewicz theorem h
2 W 2.X; /! H2.X/ is an isomorphism. (The method of the following proof can also be used to prove the classical Hurewicz theorem.) (2) Let n > 2. Assume that the theorem holds for n 1 and that ….n/ holds. We want to show that H.n/ holds and that hn W.X; /! Hn.X/ is a C-isomorphism. We ﬁrst consider the special (3) case that 2.X/ D 0 and then reduce the general case (4) to this special. (3) Thus let 2.X/ D 0. Then i.X/ Š iC1.X/ 2 C for 1 < i < n 1, and X is 1-connected. If C is saturated, then hn1 W n1.X/! Hn1.X/ is a C-isomorphism by induction. If Hi.X/ is ﬁnitely generated for i < n, then by (20.5.3) Hi.X/ is ﬁnitely generated for i < n 1 so that by induction hn1 is also an isomorphism in this case. The ﬁbration theorem shows that Hn.f / in the diagram is a C-isomorphism. From the diagram we now see that hn is a C-isomorphism. Also Hn1.X/ 2 C by induction. (4) Let now 2.X/ D 2 be arbitrary. By assumption, this group is contained in C. There exists a map W X! K.2; 2/ which induces an isomorphism 2. We pull back the path ﬁbration along X2'X PK.2; 2/ f K.2; 2/. Since 2 2 C, we have Hi.K.2; 1// 2 C for i > 0, by the general assumption in this section. Note that K D K.2; 1/ is the ﬁbre of'and f. The exact homotopy sequence of is used to show that'W i.X2/ Š 2.X/ for i > 2 and that 1.X2/ Š 0 Š 2.X2/. We can therefore apply the special case (3) to X2. Consider the diagram n.X2/
Š n.X/ hn.X2/ Hn.X2/ CŠ hn.X / Hn.X/. In order to show that hn.X/ is a C-isomorphism we show two things: (i) hn.X2/ is a C-isomorphism. (ii)'W Hn.X2/! Hn.X/ is a C-isomorphism. Part (i) follows from case (3) if we know that Hi.X2/ is ﬁnitely generated for i < n. This follows from (20.5.3) applied to the ﬁbration K! X2! X, since X is 1-connected and since 1.K/ Š 2 Š H2.X/ is ﬁnitely generated. For the proof of (ii) we ﬁrst observe that the canonical map ˇ W Hi.X2/! Hi.X2; K/ is a C-isomorphism for i > 0, since Hi.K/ 2 C for i > 0 by the 512 Chapter 20. Homology and Homotopy general assumption of this section. The ﬁbration theorem and (20.4.6) show that'W Hi.X2; K/! Hi.X; / is a C-isomorphism for 0 < i n. We now compose with ˇ and obtain (ii). (5) Now assume that H.k/ holds for 2 k < n. Since Hi.X/ 2 C for 2 i < k C 1 the Hurewicz map i.X/! Hi.X/ is a C-isomorphism for i k by H.k/, hence i.X/ 2 C for i k. By the ﬁrst part of the proof, hkC1 is a C-isomorphism, hence ….k C 1/ holds. We list some consequences of the Hurewicz theorem. Note that the general assumption of this section holds for the classes G and F. (20.6.2) Theorem. Let X be a 1-connected space. (1) i.X/ is ﬁnitely generated for i < n if and only if Hi.XI Z/ is ﬁnitely generated for i < n. (2) i.
X/ is ﬁnite for i < n if and only if (3) If X is a ﬁnite CW-complex, then its homotopy groups are ﬁnitely generated. QHi.XI Z/ is ﬁnite for i < n. (20.6.3) Theorem. Let C be a saturated Serre class. Let f W X! Y be a map between 1-connected spaces with 1-connected homotopy ﬁbre F. Then the following are equivalent: (1) k.f / W k.X/! k.Y / is a C-isomorphism for k < n and a C-epimor- phism for k D n. (2) Hk.f / W Hk.X/! Hk.Y / is a C-isomorphism for k < n and a C-epimor- phism for k D n. Proof. The statement (1) is equivalent to k.F / 2 C for k < n (exact homotopy sequence). Suppose this holds. Then Hk.X; F /! Hk.Y / is a C isomorphism for k n, by the ﬁbration theorem (20.4.1). From the exact homology sequence of the pair.X; F / we now conclude that (2) holds, since Hj.F / 2 C by the Hurewicz theorem. Here we use that 1.F / D 0. Suppose (2) holds. We show by induction that Hk.F / 2 C for k < n. Then we apply again the Hurewicz theorem. The induction starts with n D 3. Since Y and F are simply connected, the ﬁbration theorem shows that Hj.X; F /! Hj.Y; / is a C-isomorphism for j 3. The assumption (2) and the homology sequence of the pair then show H2.F / 2 C. The general induction step is of the same type. Problems 1. Let T Q be a subring. Let X be a simply connected space such that Hn.XI T / Š Hn.S nI T /. Then there exists a map S n! X which induces an isomorphism in T -homology. 20.7. Cohomology of E
ilenberg–Mac Lane Spaces 513 2. The ﬁnite CW-complex X D S 1 _ S 2 has 1.X/ Š Z. The group 2.X/ is free abelian with a countably inﬁnite number of generators. (Study the universal covering of X.) 3. Use the ﬁbration theorem and deduce for each 0-connected space a natural exact sequence 2.X/! H2.X/! H2.K.1.X/; 1/I Z/! 0. 20.7 Cohomology of Eilenberg–Mac Lane Spaces We compute the cohomology ring H.K.Z; n/I Q/. Let f W S n! K.Z; n/ D K.n/ induce an isomorphism n.f / of the n-th homotopy groups. Then also f W H n.K.Z; n/I Q/! H n.S nI Q/ is an isomorphism and n is deﬁned such that f.n/ 2 H n.S nI Q/ is a generator. (20.7.1) Theorem. If n 2 is even, then H.K.Z; n/I Q/ Š QŒn (polynomial ring). If n is odd, then f W H.K.Z; n/I Q/ Š H.S nI Q/. Proof. We work with rational cohomology. Since K.Z; 1/'S 1 and K.Z; 2/'CP 1 we know already the cohomology ring for these spaces with coefﬁcients in Z and this implies (20.7.1) in these cases. We prove the theorem by induction on n, and for this purpose we analyze the path-ﬁbration K.Z; n 1/! P! K.Z; n/ with contractible P. There are two cases for the induction step, depending on the parity of n. 2k 1 ) 2k. We have a relative ﬁbration p W.E; E 0/! K.n/ with E D K.n/I ; p.w/ D w.1/; P D E 0 D p1./. The map p W E! K.n/ is a homotopy equivalence
and E 0 is contractible. Therefore.1/ H n.E; E0/ Š H n.E/ Š H n.K.n//; the latter induced by p. The ﬁbres.F; F 0/ of p have a contractible F and F 0 D K.n/ D K.n 1/ is by induction a rational cohomology.n 1/-sphere (i.e., has the rational cohomology of S n1). Hence H k.F; F 0/ Š H k1.F 0/ Š Q for k D n and Š 0 for k 6D n. Since K.n/ is simply connected, we have a Thom class tn 2 H n.E; E0/. We can assume that tn is mapped under (1) to n, hence n is the Euler class e associated to tn. The Gysin sequence has the form (n D 2k)! H j.K.n// e! H j Cn.K.n//! H j Cn.P /! : Since P is contractible and H j.K.n// D 0 for 0 < j < n, we see inductively that the cup product with the Euler class e is an isomorphism H j.K.n//! H j C2k.K.n//. Hence H.K.n// Š QŒn. 2k ) 2k C 1. We reduce the problem to a Wang sequence. Let n D 2k C 1. We consider a pullback Y q S n P p K.n/, f 514 Chapter 20. Homology and Homotopy where f induces an isomorphism n.f /. The Wang sequence for q has the form! H j.Y / i! H j.K.n 1//! H j C1n.K.n 1//! : We use H.K.n 1// Š QŒn1 and the fact that is a derivation. From the deﬁnition of Y and the exact sequence of homotopy groups we see j.Y / D 0; j n; j.Y / Š j.S n/; j > n: From the Hurewicz theorem and the universal coefﬁcient theorem we conclude that H j.Y / D 0
for j n. Hence W H 2k.K.n 1//! H 0.K.n 1// is an isomorphism. Using the derivation property of we see inductively that W H 2kr.K.n 1//! H 2k.r1/.K.n 1// is an isomorphism, and the Wang QH.Y / D 0; and this implies QH.Y / D 0. Since Y sequence then shows us that is the homotopy ﬁbre of f, we conclude that f W H.S n/! H.K.n// is an isomorphism (by (20.4.1) say), and similarly for cohomology. This ﬁnishes the induction. 20.8 Homotopy Groups of Spheres Let n > 1 be an odd integer. Let f W S n! K.Z; n/ induce an isomorphism n.f / and denote by Y the homotopy ﬁbre of f. In the previous section we have shown that Hj.Y I Q/ D 0 for j > 0. The Hurewicz theorem modulo the class of torsion groups (D the rational Hurewicz theorem) shows us that the groups j.Y / ˝ Q are zero for j 2 N. From j.S n/ Š j.Y / for j > n we see that also j.S n/˝Q D 0 for j > n and odd n. Since we already know that the homotopy groups of spheres are ﬁnitely generated we see: (20.8.1) Theorem. Let n be an odd integer. Then the groups j.S n/ are ﬁnite for j > n. We now investigate the homotopy groups of S 2n. Let V D V2.R2nC1/ denote the Stiefel manifold of orthonormal pairs.x; y/ in R2nC1. We have a ﬁbre bundle S 2n1! V! S 2n, and V is the unit-sphere bundle of the tangent bundle of S 2n. Recall from 14.2.4 the integral homology of V Hq.V / Š q D 0; 4n 1; Z; Z=2; q D 2n 1; 0; otherwise: Let g W
V! S 4n1 be a map of degree 1. Then g induces an isomorphism in rational homology. Let F be the homotopy ﬁbre of g; it is simply connected. From (20.6.3) we see that g ˝ Q W j.V / ˝ Q! j.S 4n1/ ˝ Q is always an 20.8. Homotopy Groups of Spheres 515 isomorphism. We use (20.8.1) and see that the homotopy groups of V are ﬁnite, except 4n1.V / Š Z ˚ E, E ﬁnite. Now we go back to the ﬁbration V! S 2n and its homotopy sequence. It shows: (20.8.2) Theorem. j.S 2n/ is ﬁnite for j ¤ 2n; 4n1 and 4n1.S 2n/ Š Z˚E, E ﬁnite. The results about the homotopy groups of spheres enable us to prove a reﬁned rational Hurewicz theorem. (20.8.3) Theorem. Let X be 1-connected. Suppose Hi.XI Z/ is ﬁnite for i < k and ﬁnitely generated for i 2k 2 (k 2). Then the Hurewicz homomorphism h W r.X/! Hr.XI Z/ has ﬁnite kernel (cokernel) for r < 2k 1 (r 2k 1). Proof. The case k D 2 causes no particular problem, since the Hurewicz homomorphism h W mC1.X/! HmC1.XI Z/ is surjective for each.m 1/-connected space (m 2). So let k 3. Since X is 1-connected, the Hurewicz theorem shows that r.X/ is ﬁnite for r < k and ﬁnitely generated for r 2k 2. We write r.X/ D Fr ˚ Tr, Fr free, and Tr ﬁnite. We choose basis elements for Fr and representing maps. These representing elements provide us with a map of the form f W S D S r.1/ _
_ S r.t/! X with k r.j / 2k 2. The canonical map L j r.S r.j //! r W j S n.j / is an isomorphism for r 2k 2 and an epimorphism for r D 2k 1. We can now conclude that f W r.S/! r.X/ has ﬁnite kernel and cokernel for r 2k 2 and ﬁnite cokernel for r D 2k 1, since the homotopy groups of spheres are ﬁnite in the relevant range. The homotopy ﬁbre F of f has ﬁnite homotopy groups j.F / for j 2k 2. If, moreover, F is 1-connected, then Hj.F I Z/ is ﬁnite in the same range, by the Hurewicz theorem. The ﬁbration theorem then yields that f W Hr.S/! Hr.X/ is an F -isomorphism (F -epimorphism) for r 2k 2 (r D 2k 1). From our knowledge of the homotopy groups of spheres we see directly that the theorem holds for S. The naturality of the Hurewicz theorem applied to f is now used to see that the desired result also holds for X. We have used that F is 1-connected. This holds if 2.X/ D 0. Since 2.X/ is ﬁnite by assumption (k 3), we can pass to the 2-connected cover q W Xh2i!X of X. The map q induces F -isomorphisms in homology and homotopy. Therefore it sufﬁces to prove the theorem for Xh2i to which the reasoning above applies. If one is not interested in ﬁnite generation one obtains by a similar reasoning (see also [103]): 516 Chapter 20. Homology and Homotopy (20.8.4) Theorem. Let X be a 1-connected space. Suppose Hr.XI Q/ D 0 for r < k. Then the Hurewicz map j.X/ ˝ Q! Hj.XI Q/ is an isomorphism for j < 2k 2 and an epimorphism for j D 2k
1. This theorem indicates that homotopy theory becomes simpler “over the rationals”. In the so-called rational homotopy theory one constructs algebraic models for the rationalized homotopy theory. For an exposition see [65]. We discuss an example. Consider the path ﬁbration i! Y K.Z; 2/'K.Z; 3/! X! K.Z; 3/ D K3 with contractible X. Let f W S 3! K3 induce an isomorphism in 3 and let p! S 3 be the induced ﬁbration. The homotopy groups k.Y / are zero K2 for k 3 and p induces an isomorphism k.Y / Š k.S 3/ for k 4. (20.8.5) Proposition. 3.S 2/ Š Z and nC1.S n/ Š Z=2 for n 3. Proof. We know already that 3.S 2/ Š Z, generated by the Hopf map S 3! CP 1 Š S 2. From the Freudenthal suspension theorem we know that the suspension † W nC1.S n/! nC2.S nC1/ is surjective for n D 2 and bijective for n 3. Therefore it sufﬁces to determine 4.S 3/. By the Hurewicz theorem, 4.S 3/ Š 4.Y / Š H4.Y /. Thus it remains to compute H4.Y /. We ﬁrst determine the cohomology. (20.8.6) Proposition. The cohomology groups H k.Y / of Y are: Z for k D 0, 0 for k 0 mod 2, and Z=n for k D 2n C 1. Proof. We use K2 D K.Z; 2/ D CP 1 and H.K2/ Š ZŒc with c 2 H 2. The Wang sequence of Y! S 3 shows that H 2n.Y / Š Ker ‚; H 2nC1.Y / Š Coker ‚; since H.K2/ D 0 for odd. The group H 2n.K2/ Š Z is generated by cn. By the universal coefﬁcient formula H j.Y / D 0 for j D
1; 2; 3. Hence ‚ W H 2.K2/! H 0.K2/ is an isomorphism. We can choose c such that ‚.c/ D 1. The derivation property of ‚ yields then inductively ‚.cn/ D ncn1. The Wang sequence in homology yields in a similar manner H2n.Y / Š Coker ‚; H2nC1.Y / Š Ker ‚: From the universal coefﬁcient sequence 0! Ext.H2n.Y /; Z/! H 2nC1.Y /! Hom.H2nC1.Y /; Z/! 0 and the fact that H2n.Y / is a quotient of Z we obtain H2n.Y / Š Z=n; H2nC1.Y / D 0: The special case H4.Y / Š Z=2 now proves 4.S 3/ Š Z=2. 20.8. Homotopy Groups of Spheres 517 We apply the Hurewicz theorem modulo the class of abelian p-torsion groups (p prime) and see that the p-primary component of i.S 3/ is zero for 3 < i < 2p and isomorphic to Z=p for i D 2p. In particular an inﬁnite number of homotopy groups n.S 3/ is non-zero. The determination of the homotopy groups of spheres is a difﬁcult problem. You can get an impression by looking into [163]. Individual computations are no longer so interesting; general structural insight is still missing. Since the groups nCk.S n/ do not change after suspension for k n 2, by the Freudenthal theorem, they are called the stable homotopy groups S k. We copy a table from [185]; a denotes a cyclic group of order a, and a b is the product of cyclic groups of order a and b, and aj the j -fold product of cyclic groups of order a. k 0 1 S k 1 2 11 10 k 2 2 3 24 4 0 5 0 6 2 12 13 14 15 16 S k 6 504 0 3 22 480 2 22 7 240 17 24 8 22 18 9 23 19 8 2 264 2 20.8.7 The Hopf invariant
. The exceptional case 4n1.S 2n/ is interesting in many respects. Already Hopf constructed a homomorphism h W 4n1.S 2n/! Z, now called the Hopf invariant, and gave a geometric interpretation in the simplicial setting [88]. Let f W S 4n1! S 2n be a smooth map; and let a, b be two regular values. The pre-images Ma D f 1.a/ and Mb D f 1.b/ are closed orientable.2n 1/-manifolds, they have a linking number, and this number is the Hopf invariant of f. It is easy to deﬁne h.f /, using cohomology. Let f W S 2k1! S k be given.k 2/. Attach a 2k-cell to S k by f and call the result X D X.f /. The inclusion i W S k! X induces an isomorphism H k.X/ Š H k.S k/, and we also have an isomorphism H 2k.D2k; S 2k1/ Š H 2k.X; S k/! H 2k.X/. The integral cohomology groups H j.X/ for j 6D 0; k; 2k are zero. Choose generators x 2 H k.X/ and y 2 H 2k.X/. Then there holds a relation x Y x D h.f /y in the cohomology ring. The graded commutativity of the cup product is used to show that h.f / D 0 for odd k. In the case k D 2n the integer h.f / is the Hopf invariant. Since X.f / depends up to h-equivalence only on the homotopy class of f, the integer h.f / is a homotopy invariant. One shows the elementary properties of this invariant: (1) h is a homomorphism. (2) If g W S 4n1! S 4n1 has degree d, then h.fg/ D dh.f /. (3) If k W S 2n! S 2n has degree d, then h.kf / D d 2h.f /. 518 Chapter 20. Homology and Homotopy Note that a map of degree d does not induce the multiplication by d, as
opposed to the general situation for cohomology theories. It is an important problem to determine the image of h. Already Hopf showed by an explicit construction that 2Z is always contained in the image. Here is the Hopf construction. Start with a map u W S k S k! S k. From it we obtain a map f W S 2kC1! S kC1 via the diagram uid f S k I p †S k, where q is the projection onto the suspension and p the projection onto the join. The map u has a bi-degree.a; b/. Hopf shows that (with suitable orientations chosen) h.f / D ab. The map S 2n1 S 2n1! S 2n1,.x; / 7! hx; ix has bidegree.2; 1/. If ˛ W Rn Rn! Rn is a bilinear map without zero divisors (i.e., ˛.x; y/ D 0 implies that x or y is zero), then.Rn; ˛/ is called an n-dimensional real division algebra. The induced map ˇ W S n1 S n1! S n1,.x; y/ 7! ˛.x; y/=k˛.x; y/k satisﬁes ˇ."x; y/ D "ˇ.x; y/ for "; 2 f˙1g. Hence ˇ has a bi-degree.d1; d2/ with odd dj. If there exist maps with odd Hopf invariant, then there also exist maps with invariant 1, since 2Z is contained in the image of h. It is a famous result of Adams [2] that maps f W S 4n1! S 2n of Hopf invariant 1 only exist for n D 1; 2; 4. Hence there exist n-dimensional real division algebras only for n D 1; 2; 4; 8. See [55] for this topic and the classical algebra related to it. Once complex K-theory is established as a cohomology theory, it is fairly easy Þ to solve the Hopf invariant 1 problem [5]. 20.9 Rational Homology Theories Recall the n-th stable homotopy group!n.X/ D colimk n
Ck.X ^ S.k// of the pointed space X. The Hurewicz homomorphisms are compatible with suspension, i.e., the diagram n.Y / † h QHn.Y / † nC1.†Y / n QHnC1.†Y / is commutative for each well-pointed space Y. This follows from the inductive deﬁnition of the Hurewicz homomorphisms; one has to use the same deﬁnition of † in homotopy and homology via the boundary operator of the pair.C Y; Y / and the quotient map C Y! C Y =Y D †Y. We pass to the colimit and obtain the 20.9. Rational Homology Theories 519 stable Hurewicz homomorphism hn W!n.X/! QHn.X/: The hn are natural transformations of homology theories (on well-pointed spaces). In order to see this one has to verify that the diagram!n.X/ hn QHn.X/!nC1.X ^ S.1// hnC1 † QHnC1.X ^ S.1// is commutative. The commutativity of this diagram is a reason for introducing the sign in the deﬁnition D.1/nl of the suspension isomorphism for spectral homology. The coefﬁcients of the theory!./ are the stable homotopy groups of spheres!n.S 0/ D colimk nCk.S k/. These groups are ﬁnite for n > 0. Finite abelian groups become zero when tensored with the rational numbers. We thus obtain a natural transformation of homology theories h W!.X/ ˝Z Q! QH.X/ ˝Z Q Š QH.XI Q/ which are isomorphisms on the coefﬁcients and therefore in general for pointed CW-complexes. This basic rational isomorphism is now used to show that any homology theory Qh with values in Q- MOD can be reduced to ordinary homology. We deﬁne natural maps!p.X/ ˝Z Qhq.S 0/! QhpCq.X/: Let
x 2!p.X/ be represented by f W S.p C k/! X ^ S.k/. The image of y 2 Qhq.S 0/ under Qhq.S 0/ Š QhqCpCk.S.p C k// f! QhqCpCk.X ^ S.k// Š QhpCq.X/ L is independent of the chosen representative f of x. We combine these homomorphisms pCqDn!p.X/ ˝Z Qhq.S 0/! Qhn.X/ and obtain a natural transformation of homology theories. Now assume that the coefﬁcients Qhq.S 0/ are Q-vector spaces. Then for X D S 0 only the groups!0.S 0/ ˝Z Qhn.S 0/ are non-zero; and the induced map to Qhn.S 0/ is an isomorphism, since!0.S 0/ Š Z by the degree and a map of degree k induces on Qhn.S 0/ the multiplication by k. We thus have shown: 520 Chapter 20. Homology and Homotopy (20.9.1) Theorem. Let Qh be an additive homology theory for pointed CW-complexes with values in Q-vector spaces. Then we have an isomorphism L pCqDn QHp.XI Qhq.S 0// Š L pCqDn QHp.X/ ˝ Qhq.S 0/ Š Qhn.X/ Š Š L pCqDn!p.X/ ˝ Qhq.S 0/ of homology theories. If Qk./ is an arbitrary additive homology theory we can apply the foregoing to Qh./ D Qk./ ˝Z Q. Problems 1. The Eilenberg–Mac Lane spectrum.K.Z; n/ j n 2 N0/ yields a homology theory which is isomorphic to singular homology with integral coefﬁcients. 2. One can deﬁne the stable Hurewicz transformation from a morphism of the sphere spectrum into the Eilenberg–Mac Lane spectrum which is obtained from maps S n! K.Z; n/ that induce isomorphisms of n. 3. De
ﬁne a stable Hurewicz homomorphism hn W!n.X/! QH n.XI Z/ either from a map of spectra or by an application of cohomology to representing maps of elements in!n.X/. Construct a natural commutative diagram!n.X/ hn QH n.X/ Hom.!n.X/; Z/.hn/ Hom. QHn.X/; Z/. 4. Give a proof of the (absolute) Hurewicz theorem by using the K.Z; n/-deﬁnition of homology. The proof uses: Let j.X/ D 0 for j < n 2; then j.X ^ K.Z; k// D 0 for j n C k 1 and nCk.X ^ K.Z; k// Š n.X/. 5. Derive an isomorphism Qhn.X/ Š with values in Q-vector spaces (X a ﬁnite pointed CW-complex). 6. Use a ﬁbration K.Z; n/! K.Z; n/! K.Z=k; n/ and derive a universal coefﬁcient sequence for homology with Z=k-coefﬁcients. 7. An interesting example of a rational cohomology isomorphism is given by the Chern character. It is a natural isomorphism of Q-algebras QH p.XI hq.S 0// for cohomology theories pCqDn Q ch W K.X/ ˝ Q! Q n H 2n.XI Q/; for ﬁnite complexes X, say, and which sends a complex line bundle over X to the power series P1 ec1./ D iD0 1 iŠ c1./i 2 H 2.XI Q/: Chapter 21 Bordism We begin with the deﬁnition of bordism homology. The geometric idea of homology is perhaps best understood from the view-point of bordism and manifolds. A “singular” cycle is a map from a closed manifold to a space, and the boundary relation is induced by manifolds with boundary. Several of our earlier applications of homology and homotopy can easily be obtained just from the existence of b
ordism homology, e.g., the Brouwer ﬁxed point theorem, the generalized Jordan separation theorem and the component theorem, and the theorem of Borsuk–Ulam. Bordism theory began with the fundamental work of Thom [184]. He determined the bordism ring of unoriented manifolds (the coefﬁcient ring of the associated bordism homology theory). This computation was based on a fundamental relation between bordism and homotopy theory, the theorem of Pontrjagin–Thom. In the chapter on smooth manifolds we developed the material which we need for the present proof of this theorem. One application of this theorem is the isomorphism between the geometric bordism theory and a spectral homology theory via the Thom spectrum. From this reduction to homotopy we compute the rational oriented bordism. Hirzebruch used this computation in the proof of his signature theorem. This proof uses almost everything that we developed in this text. 21.1 Bordism Homology We deﬁne the bordism relation and construct the bordism homology theory. Manifolds are smooth. Let X be a topological space. An n-dimensional singular manifold in X is a pair.B; F / which consists of a compact n-dimensional manifold B and a continuous map F W B! X. The singular manifold @.B; F / D.@B; F j@B/ is the boundary of.B; F /. If @B D ;, then.B; F / is closed. A null bordism of the closed singular manifold.M; f / in X is a triple.B; F; '/ which consists of a singular manifold.B; F / in X and a diffeomorphism'W M! @B such that.F j@B/ı' D f. If a null bordism exists, then.M; f / is null bordant. Let.M1; f1/ and.M2; f2/ be singular manifolds in X of the same dimension. We denote by.M1; f1/ C.M2; f2/ the singular manifold hf1; f2 i W M1 C M2! X. We say.M1; f1/ and.M2; f2/ are bordant, if.M1; f1/ C.M2
; f2/ is null bordant. A null bordism of.M1; f1/C.M2; f2/ is called bordism between.M1; f1/ and.M2; f2/. The boundary @B of a bordism.B; F; '/ between.M1; f1/ and.M2; f2/ thus consists of a disjoint sum @1B C @2B, and'decomposes into two diffeomorphisms 'i W Mi! @i B. 522 Chapter 21. Bordism (21.1.1) Proposition. “Bordant” is an equivalence relation. Proof. Let.M; f / be given. Set B D M I and F D f ıpr W M I! M! X. Then @B D M 0 C M 1 is canonically diffeomorphic to M C M and.B; F / is a bordism between.M; f / and.M; f /. The symmetry of the relation is a direct consequence of the deﬁnition. Let.B; F; 'i W Mi! @i B/ be a bordism between.M1; f1/ and.M2; f2/ and.C; G; i W Mi! @i C / a bordism between.M2; f2/ and.M3; f3/. We identify in B C C the subset @2B with @2C via x 2'1 2.x/ for x 2 @2B. The result D carries a smooth structure, and the canonical maps B! D C are smooth embeddings (15.10.1). We can factor hF; G i W B C C! X over the quotient map B C C! D and get H W D! X, and.D; H; h'1; 3 i/ is a bordism between.M1; f1/ and.M3; f3/. We denote by ŒM; f the bordism class of.M; f / and by Nn.X/ the set of bordism classes of n-dimensional closed singular manifolds in X. The set Nn.X/ carries an associative and commutative composition law ŒM; f C ŒN; g D ŒM C
N; hf; g i. The reader may check that this is well-deﬁned. (21.1.2) Proposition..Nn.X/; C/ is an abelian group. Each element has order at most 2. Proof. The class of a null bordant manifold serves as neutral element, for example the constant map S n! X. (For the purpose at hand it is convenient to think of the empty set as an n-dimensional manifold.) For each.M; f / the sum.M C M; hf; f i/ is null bordant, hence ŒM; f C ŒM; f D 0. A continuous map f W X! Y induces a homomorphism Nn.f / D f W Nn.X/! Nn.Y /; ŒM; g 7! ŒM; fg: In this way Nn./ becomes a functor from TOP to ABEL. Homotopic maps induce the same homomorphism: If is a homotopy, then.M I; F ı.hid// is a bordism between.M; f h/ and.M; gh/. If X is empty, we consider Nn.X/ as the trivial group. (21.1.3) Example. A 0-dimensional compact manifold M is a ﬁnite discrete set. Hence.M; f / can be viewed as a family.x1; : : : ; xr / of points in X. Points x; y 2 X are bordant if and only if they are contained in the same path component. (Here you have to know 1-dimensional compact manifolds.) One concludes that N0.X/ is isomorphic to the Z=2-vector space over 0.X/. Þ (21.1.4) Proposition. Let h W K! L be a diffeomorphism. Then ŒL; g D ŒK; gh. Proof. Consider the bordism g ı pr W L I! X; on the boundary piece L 1 we use the canonical diffeomorphism to L, on the boundary piece we L 0 we compose the canonical diffeomorphism to L with h. 21.1. Bordism Homology 523 We now make the functors Nn./ part of a homology theory. But this time, for variety, we
do not begin with the deﬁnition of relative homology groups. The exact homology sequence and the excision axiom are now replaced by a Mayer–Vietoris sequence. Suppose X is the union of open sets X0 and X1. We construct the boundary operator @ W Nn.X/! Nn1.X0 \ X1/ of the Mayer–Vietoris sequence. Let ŒM; f 2 Nn.X/ be given. The sets Mi D f 1.X X Xi / are disjoint closed subsets of M. (21.1.5) Lemma. There exists a smooth function ˛ W M! Œ0; 1 such that: (1) Mi ˛1.i/ for i 2 f0; 1g. (2) 1 2 is a regular value of ˛. We call ˛ in (21.1.5) aseparating function. If ˛ is a separating function, then 2 / is a closed submanifold of M of dimension n 1 (or empty), and M˛ D ˛1. 1 f induces by restriction f˛ W M˛! X0 \ X1. If t ¤ 0; 1 is another regular value of ˛, then ˛1.t/ and ˛1. 1 via ˛1Œ 1 being given by any choice of a regular value t 2 0; 1Œ of ˛ with M˛ D ˛1.t/. 2 / are bordant 2 is therefore immaterial. We think of ŒM˛; f˛ as 2 ; t. The choice of 1 (21.1.6) Lemma. Let ŒK; f D ŒL; g 2 Nn.X/ and let ˛; ˇ be separating functions for.K; f /;.L; g/. Then ŒK˛; f˛ D ŒLˇ ; gˇ. Proof. We take advantage of (21.1.4). Let.B; F / be a bordism between.K; f / and.L; g/ with @B D K C L. There exists a smooth function W B! Œ0; 1 such that jK D ˛; jL D ˇ;
F 1.X X Xj / 1.j /: We choose a regular value t for and j@B and obtain a bordism 1.t/ between some K˛ and some Lˇ. From (21.1.6) we obtain a well-deﬁned boundary homomorphism @ W Nn.X/! Nn1.X0 \ X1/; ŒM; f 7! ŒM˛; f˛: (21.1.7) Proposition. Let X be the union of open subspaces X0 and X1. Then the sequence @! Nn.X0 \ X1/ j! Nn.X0/ ˚ Nn.X1/ k! Nn.X/ @! is exact. Here j.x/ D.j 0 j W X0 \ X1! X and k W X! X. The sequence ends with.x// and k.y; z/ D k0.x/; j 1 y k1 z with the inclusions k! N0.X/! 0. 524 Chapter 21. Bordism Proof. (1) Exactness at Nn1.X0 \ X1/. Suppose ŒM; f 2 Nn.X/ is given. Then M is decomposed by M˛ into the parts B0 D ˛1Œ0; 1 2 ; 1 with common boundary M˛. Since f.B0/ X1, we see via B0 that j 1 @ŒM; f is in Nn1.X1/ the zero element. This shows j ı @ D 0. 2 and B1 D ˛1Œ 1 Suppose, conversely, that j ŒK; f D 0. Then there exist singular manifolds.Bi ; Fi / in Xi such that @B0 D K D @B1 and F0jK D f D F1jK. We identify B0 and B1 along K and obtain M ; the maps F0 and F1 can be combined to F W M! X. There exists a separating function ˛ on M such that M˛ D K: With collars of K in B0 and B1 one obtains an embedding K Œ0; 1 M which is the identity on K f 1 4. By construction, @ŒM;
F D ŒK; f. 2 g; then one chooses ˛ such that ˛.k; t/ D t for k 2 K, 1 4 t 3 (2) Exactness at Nn.X0/ ˚ Nn.X1/. By deﬁnition, k ı j D 0. Suppose xi D ŒMi ; fi 2 Nn.Xi / are given. If k.x0; x1/ D 0 there exists a bordism.B; F / in X between.M0; k0f0/ and.M1; k1f1/. Choose a smooth function W B! Œ0; 1 such that: (1) F 1.X X X1i / [ Mi 1.i/ for i D 0; 1. (2) has regular value 1 2. 2 /; F j 1. 1 Let.N; f / D. 1. 1 2 / is a bordism between.N; f / and.M0; f0/ in X0; similarly for.M1; f1/. This shows j ŒN; f D.x0; x1/. 2 //. Then. 1Œ0; 1 2 ; F j 1Œ0; 1 (3) Exactness at Nn.X/. The relation @ ı k D 0 holds, since we can choose on.M0; k0f0/ C.M1; k1f1/ a separating function ˛ W M0 C M1! Œ0; 1 such that ˛1. 1 2 / is empty. 2 and B0 D ˛1Œ 1 Conversely, let ˛ be a separating function for.M; f / in X and.B; F / a null bordism of.M˛; f˛/ in X0 \ X1. We decompose M along M˛ into the manifolds B1 D ˛1Œ0; 1 2 ; 1 with @B1 D M˛ D @B0. Then we identify B and B0 along M˛ D K and obtain a singular manifold.M0; f0/ D.B0 [K B;.f jB0/ [K F / in X0, and similarly.M1; f1/ in X1.
Once we have shown that in Nn.X/ the equality ŒM0; f0 C ŒM1; f1 D ŒM; f holds, we have veriﬁed the exactness. We identify in M0 I CM1 I the parts B 1 in M0 1 and M1 1. The resulting manifold L D.M0 I /[B1.M1 I / has the boundary.M0 CM1/CM. A suitable map F W L! X is induced by.f0; f1/ ı pr1 W.M0 C M1/ I! X. For the smooth structure on L see 15.10.3. We now deﬁne relative bordism groups Nn.X; A/ for pairs.X; A/. Elements of Nn.X; A/ are represented by maps f W.M; @M /!.X; A/ from a compact nmanifold M. Again we call.M; f / D.M; @M I f / a singular manifold in.X; A/. The bordism relation is a little more complicated. A bordism between.M0; f0/ and.M1; f1/ is a pair.B; F / with the following properties: (1) B is a compact.n C 1/-manifold with boundary. (2) @B is the union of three submanifolds with boundary M0, M1 and M 0, where @M 0 D @M0 C @M1; Mi \ M 0 D @Mi. 21.1. Bordism Homology 525 (3) F jMi D fi. (4) F.M 0/ A. We call.M0; f0/ and.M1; f1/ bordant, if there exists a bordism between them. Again “bordant” is an equivalence relation. For the proof one uses 15.10.3. The sum in Nn.X; A/ is again induced by disjoint union. Each element in Nn.X; A/ has order at most 2. A continuous map f W.X; A/!.Y; B/ induces a homomorphism Nn.f / D f W Nn.X; A/! Nn.Y; B/ by composition with f. If
f0 and f1 are homotopic as maps between pairs, then Nn.f0/ D Nn.f1/. The assignment ŒM; f 7! Œ@M; f j@M induces a homomorphism (boundary operator) @ W Nn.X; A/! Nn1.A/. For A D ; the equality Nn.X; ;/ D Nn.X/ holds. (21.1.8) Lemma. Let M be a closed n-manifold and V M an n-dimensional submanifold with boundary. If f W M! X is a map which sends M X V into A, then ŒM; f D ŒV; f jV in Nn.X; A/. Proof. Consider F W M I! X,.x; t/ 7! f.x/. Then @.M I / D M @I and V 1 [ M 0 is a submanifold of @.M I / whose complement is mapped under F into A. The deﬁnition of the bordism relation now yields the claim. (21.1.9) Proposition. Let i W A X and j W X D.X; ;/.X; A/. Then the sequence @! Nn.A/ i! Nn.X/ j! Nn.X; A/ @! is exact. The sequence ends with j! N0.X; A/! 0. Proof. (1) Exactness at Nn.A/. The relation i ı @ D 0 is a direct consequence of the deﬁnitions. Let.B; F / be a null bordism of f W M! A in X. Then @ŒB; F D ŒM; f. (2) Exactness at Nn.X/. Let ŒM; f 2 Nn.A/ be given. Choose V D ; in (21.1.8). Then ŒM; f D 0 in Nn.X; A/, and this shows ji D 0. Let jŒM; f D 0. A null bordism of ŒM; f in.X; A/ is a bordism of.M; f / in X to.K; g/ such that g.K/ A. A bordism of
this type shows iŒK; g D ŒM; f. (3) Exactness at Nn.X; A/. The relation @ıi D 0 is a direct consequence of the deﬁnitions. Let @ŒM; f D 0. Choose a null bordism ŒB; F of.@M; f j@M /. We identify.M; f / and.B; f / along @M and obtain.C; g/. Lemma (21.1.8) shows jŒC; g D ŒM; f. A basic property of the relative groups is the excision property. It is possible to give a proof with singular manifolds. (21.1.10) Proposition. The inclusion i W.X X U; A X U /!.X; A/ induces an isomorphism i W Nn.X X U; A X U / Š Nn.X; A/, provided xU Aı. 526 Chapter 21. Bordism The bordism notion can be adapted to manifolds with additional structure. Interesting are oriented manifolds. Let M0 and M1 be closed oriented n-manifolds. An oriented bordism between M0; M1 is a smooth compact oriented.n C 1/-manifold B with oriented boundary @B together with an orientation preserving diffeomorphism'W M1 M0! @B. Here we have to use the convention about the boundary orientation, and M1 M0 denotes the disjoint sum of the manifolds M1 and M0 where M1 carries the given and M0 the opposite orientation. Again this notion of bordism is an equivalence relation. Singular manifolds are deﬁned as before, and we have bordism groups n.X/ of oriented bordism classes of singular n-manifolds in X. But now elements in the bordism group no longer have order at most 2. For a point P we have 0.P / D Z, i.P / D 0 for 1 i 3. The assertion about 1 follows from the fact that S 1 is an oriented boundary; the known classiﬁcation of orientable surfaces as a sphere with handles shows that these surfaces are oriented boundaries. It is a remarkable result that 3.P / D 0: Each oriented closed 3-manifold is an oriented boundary
; for a proof of this theorem of Rohlin see [77]. The exact sequences (21.1.7) and (21.1.9) as well as (21.1.10) still hold for the -groups. The deﬁnition of the boundary operator @ W n.X; A/! n1.A/ uses the boundary orientation. In order to deﬁne the boundary operator of the MV-sequence we have to orient M˛. There exists an open neighbourhood U of M˛ in M and a diffeomorphism'W V D 1=2 "; 1=2 C "Œ M˛! U such that.˛'/.t; x/ D t. If M is oriented, we have the induced orientation of U, and we orient V such that'preserves the orientation. We orient M˛ such that V carries the product orientation. The idea of bordism can be used to acquire an intuitive understanding of homology. A compact n-manifold M has a fundamental class zM 2 Hn.M; @M I F2/ and @zM 2 Hn1.@M I F2/ is again a fundamental class. Let f W.M; @M /!.X; A/ be a singular n-manifold. We set.f / D fzM 2 Hn.X; AI F2/. In this manner we obtain a well-deﬁned homomorphism W Nn.X; A/! Hn.X; AI F2/: The morphisms constitute a natural transformation of homology theories. One of the basic results of bordism theory says that is always surjective. This allows us to view homology classes as being represented by singular manifolds. If, in particular, f is an embedding of manifolds, then we view the image of f as a cycle or a homology class. In bordism theory, the fundamental class of M is M itself, i.e., the identity of M considered as a singular manifold. The transformation can be improved if we take tangent bundle information into account. Let M be a compact n-manifold and denote by M W M! BO the classifying map of the stable tangent bundle of M. For a singular manifold f W.M; @M /!.X; A/ we then have.f;
M / W.M; @M /!.X; A/ BO. Again we take the image of the fundamental class Œf D.f; M /zM 2 Hn..X; A/ BOI F2/: 21.1. Bordism Homology 527 We now obtain a natural transformation of homology theories D.X; A/ W N.X; A/! H..X; A/ BOI F2/; and in particular for the coefﬁcient ring, the Thom bordism ring N of unoriented manifolds, W N! H.BOI F2/: A fundamental result says that.X; A/ is always injective [28, p. 185]. This transformation is also compatible with the multiplicative structures. The algebra structure of H.X BOI F2/ is induced by the homology product and the H -space structure m W BO BO! BO which comes from the Whitney sum of bundles. We obtain a natural homomorphism of graded algebras, H.X BOI F2/ ˝ H.Y BOI F2/! H.X BO Y BOI F2/! H.X Y BOI F2/I the ﬁrst map is the homology product and the second is induced by the permutation of factors and m. Thom [184] determined the structure of the ring N: It is a graded algebra F2Œu2; u4; u5; : : : with a generator uk in each dimension k which does not have the form k D 2t 1. One can take u2n D ŒRP 2n as generators in even dimensions. The ring H.BOI F2/ is isomorphic to F2Œa1; a2; a3; : : : with a generator ai in dimension i. Another basic result says that there exists a natural isomorphism N.X/ Š N ˝F2 H.XI F2/ of multiplicative homology theories [160], [28, p. 185]. Thus the homology theory N./ can be reduced to the determination of the coefﬁcient ring N and singular homology with F2-coefﬁcients. For oriented manifolds the situation is more complicated. One can still deﬁne a multiplicative natural transformation of homology theories W.
X; A/! H..X; A/ BSOI Z/ from the fundamental classes of oriented manifolds as above. But this time the transformation is no longer injective and.X; A/! H.X; AI Z/ in general not surjective. Also the theory./ cannot be reduced to ordinary homology. But the transformation still carries a lot of information. It induces a natural isomorphism.X; A/ ˝ Q Š H..X; A/ BSOI Q/; and, in particular,.1/ ˝ Q Š H.BSOI Q/ by the stable classifying map of the tangent bundle (see (21.4.2)). The ring ˝ Q is isomorphic to QŒx4; x8; : : : with a generator xn for each n 0.4/. One can take the x4n D ŒCP 2n as polynomial generators (see (21.4.4)). 528 Chapter 21. Bordism We have seen that the signature of oriented smooth manifolds deﬁnes a homomorphism W ˝ Q! Q of Q-algebras with ŒCP 2n D 1. The isomorphism (1) tells us that this homomorphism can be determined from the stable tangent bundle. A famous formula of Hirzebruch [81], the so-called Hirzebruch L-genus, gives a polynomial Ln.p1; p2; : : : ; pn/ in the Pontrjagin classes p1; : : : ; pn such that the evaluation on the fundamental class of an oriented 4n-manifold is the signature hLn.p1.M /; : : : ; pn/; ŒM i D.M /: It is a remarkable fact that the polynomials Ln have rational coefﬁcients with large denominators, but nevertheless the evaluation on the fundamental class is an integer. Such integrality theorems have found a conceptual interpretation in the index theory of Atiyah and Singer [16]. The homomorphisms Nn! Hn.BOI F2/ and n! Hn.BSOI Z/ have an interpretation in terms of characteristic numbers and can be determined by evaluating polynomials in the Stiefel–Whitney classes or Pontr
opy class of g" is independent of the chosen (sufﬁciently small) ", by a linear homotopy in the ﬁbre. Let us set P.M; F / D Œg" 2 ŒQc; M./0. We say that a map g" is obtained by a Pontrjagin–Thom construction. (21.2.1) Lemma. For -bordant manifolds.Mj ; Fj / the classes P.Mj ; Fj / are equal. Therefore we obtain a well-deﬁned map P W L.Q; /! ŒQc; M./0. Proof. We apply a Pontrjagin–Thom construction to a -bordism. For this purpose we use for Q I the product embedding. Let.W; F / be a -bordism. We obtain a tubular map W E.W /! Q I which is over M0 Œ0; 1=3Œ the product of the tubular map for M0 with Œ0; 1=3Œ, and similarly for the other end. For sufﬁciently small ", again embeds E2".W / onto a neighbourhood of W and we deﬁne as above a Pontrjagin–Thom map Qc I! Qc I =.Qc I X U"/! D".W /=S".W /! M.W /! M./; and this map is a homotopy between the Pontrjagin–Thom maps for.M0; F0/ and.M1; F1/. (21.2.2) Theorem (Pontrjagin, Thom). The Pontrjagin–Thom map P W L.Q; /! ŒQc; M./0 is a bijection. Proof. We construct a map in the other direction. Let us observe that the maps f W Qc! M./ obtained by the Pontrjagin–Thom construction are of a very special type. They have the following properties: (1) The map f W f 1.E.//! E./ is proper, smooth and transverse to the zero section B E./. 21.2. The Theorem of Pontrjagin and Thom 531 (2) There exists a tubular neighbourhood U" of M D f 1.B/
in Q such that f.x/ D 1,x … U". (3) The map f ı ı h1 " W E.M /! E".M /! U"! E./ is a smooth bundle map. So we have to deform an arbitrary map into one having these three properties. (i) Let g W Qc! M./ be given and set A D g1.E.//. Then A is open in Q and g W A! E./ is a proper map. By the approximation theorem (15.8.4) there exists a proper homotopy of g to a smooth map g1 W A! E./. Restrict g1 to a compact neighbourhood V of g1 1.B/ in A such that V is a manifold and g1.@V / E./ X B. By the transversality theorem (15.9.8) we ﬁnd a smooth homotopy of g1jV to a map which is transverse to B and such that the homotopy is constant in a neighbourhood of @V. We can therefore extend this homotopy to a smooth proper homotopy of g1 by a constant homotopy in the complement of V. Since both homotopies are proper, they can be extended continuously to Qc by mapping the complement of A to the base point. The result is a smooth map g2 W Qc! M./ which has property (1) above. (ii) Let M D g1 2.B/. Let now " be small enough such that the tubular neighbourhood U2" of M is contained in A. Let ˇ W Qc! Œ0; 1 be a continuous function which is smooth on Q and such that ˇ1.0/ D D"=2 and ˇ1Œ0; 1Œ D U". We deﬁne a homotopy of g2 by ( Ht.x/ D.1 tˇ.x//1 g2.x/; x 2 A; t < 1I x 2 U"; t D 1; 1; otherwise: The map g3 D H1 has properties (1) and (2) above with U D U". (iii) Let f D g3 be a map obtained in step (ii). Consider the composition f h1 " D h W E.M /! E".M /! U"!
E./: This map is proper, smooth, and transverse to B E./. The homotopy Ht.x/ D t 1h.tx/, deﬁned for t > 0 can be extended to t D 0 by a bundle map ˆ W M! such that the resulting homotopy is smooth and proper. The map ˆ is the derivative in the direction of the ﬁbres. In order to see what happens in the limit t! 0, we express h in local coordinates. Then h has the form X Rn! Y Rn;.x; v/ 7!.a.x; v/; b.x; v// with open sets X M, Y B and a.x; 0/ D f.x/; b.x; 0/ D 0. Then Ht.x; v/ D.a.x; tv/; t 1b.x; tv//. The map v 7! limt!0 t 1b.x; tv/ is the differential of bx W v 7! B.x; v/ at v D 0. It is a bijective linear map, because f is transverse to the zero section. (iv) We now construct a map Q which is inverse to P. Suppose g W Qc! M./ is a map such that g W g1E./! E./ is proper, smooth, and transverse to the 532 Chapter 21. Bordism zero section. Let M D g1.B/. Then the differential of g induces a smooth bundle map F W M!. The map Q sends Œg to the bordism class of the -submanifold.M; F /. By (i) we know that each homotopy class has a representative g with the properties just used. If g0 and g1 are two such representatives, we choose a homotopy between them which is constant on Œ0; 1=3Œ and on 2=3; 1 and apply the method of (i) in order to obtain a homotopy h W Qc I! M./; g0'g1 such that h W h1E./! E./ is proper, smooth, transverse to the zero section, and constant on Œ0; Œ and 1 ; 1. The pre-image of the zero section and the differential of h yield a -bordism
. This argument shows that Q is well-deﬁned. By construction, QP is the identity. The arguments of (i)–(iii) show that P is surjective. (21.2.3) Example. Let Q D M be a closed connected n-manifold and the ndimensional bundle over a point. Then E./ D Rn and M./ D S n D Rn[f1g. A -submanifold of Q is a ﬁnite subset X together with an isomorphism Fx W TxM! Rn for each x 2 X. In the present situation ŒQc; M./0 D ŒM; S n. We are therefore in the situation of the Hopf degree theorem. P Let M be oriented. If Fx is orientation preserving, we set ".x/ D 1, and x2X ".x/ 2 Z. The integer ".X; F / ".x/ D 1 otherwise. Let ".X; F / D characterizes the -bordism class. If we represent a homotopy class in ŒM; S n by a smooth map f W M! S n with regular value 0 2 Rn, then X D f 1.0/ and Fx D Txf and ".X; F / is the degree of f, as we have explained earlier. If we show that ".X; F / characterizes the bordism class, then the Pontrjagin–Thom theorem gives a proof of the Hopf degree theorem for smooth manifolds. If M is non-orientable, we have a similar situation, but this time we have to Þ consider ".X; F / modulo 2. (21.2.4) Example. Let Q D RnCk and E./ D Rn. Then ŒQc; M./0 D nCk.S n/, but we disregard the group structure for the moment. A -submanifold is in this case a closed k-manifold M RkCn together with a trivialization of the normal bundle. A trivialization of a vector bundle is also called a framing of the bundle. Since the normal bundle is inverse to the tangent bundle we see that the tangent bundle is stably trivial and a framing! n" of the normal bundle induces TM ˚ n"! TM ˚!.n C k/",
a stable framing of M. We denote by!n.k/ the bordism set of a closed n-manifold with framing TM ˚ k"!.n C k/". The bordism relation is deﬁned as follows. Let W be a bordism between M0 and M1 and let ˆ W T W ˚.k 1/"!.n C k/" be a framing. Let 0 W T W jMo Š TM 0 ˚ " where the positive part of " corresponds to an inwards pointing vector. Similarly 1 W T W jM1 Š TM1 ˚ " where now the positive part corresponds to an outwards pointing vector. We obtain a framing 'i W TM i ˚ " ˚.k 1/"! T W jMi ˚.k 1/" ˆ!.n C k/" of Mi. We say in this case:.W; ˆ/ is a framed bordism between.M0; '0/ and.M1; '1/. 21.2. The Theorem of Pontrjagin and Thom 533 The assignment ŒM; '! ŒM;'is a well-deﬁned map L.RnCk; k"/! Þ!n.k/. It is bijective for k > n C 1. In the Pontrjagin–Thom theorem it is not necessary to assume that is a smooth bundle over a closed manifold. In fact, can be an arbitrary bundle. In this case we have to deﬁne the Thom space in a different manner. Let have a Riemannian metric; then we have the unit disk bundle D./ and the unit sphere bundle S./. We deﬁne the Thom space now as the quotient space M./ D D./=S./. A deﬁnition that does not use the Riemannian metric runs as follows. The multiplicative group R C of positive real numbers acts on the subset E0./ of non-zero vectors ﬁbrewise by scalar multiplication. Let S./ be the orbit space with induced projection s W S./! B. The mapping cylinder of s is a space d W D./! B over B and M./ is deﬁned to be the (unpointed) mapping cone of s.
From this deﬁnition we see that a bundle map f W! induces a pointed map M.f / W M./! M./. In the category of compactly generated spaces we have a canonical homeomorphism M. / Š M./ ^ M./. If is the trivial one-dimensional bundle over a point, this homeomorphism amounts to M. ˚ "/ Š M./ ^ S.1/ with S.1/ D R [ f1g. We can now deﬁne as before -submanifolds of Q and -bordisms. Also the Pontrjagin–Thom construction can be applied in this situation, and we obtain a well-deﬁned map P D P W L.Q; /! ŒQc; M./0: These maps constitute a natural transformation between functors from the category of n-dimensional bundles and bundle maps. (21.2.5) Theorem (Pontrjagin, Thom). The Pontrjagin–Thom map P is for each bundle a bijection. Proof. The proof is essentially a formal consequence of the special case (21.2.2), based on the general techniques developed so far. (i) In the proof of (21.2.2) we used smooth bundles. A bundle over a closed manifold B is induced from a tautological bundle over some Grassmannian Gn.RN / by some map f. The map f is homotopic to a smooth map g and the bundle induced by g is therefore smooth and isomorphic to. This fact allows us to work with arbitrary bundles in (21.2.2). The Pontrjagin–Thom construction itself does not use a smooth bundle map. (ii) Suppose i W X! Y and r W Y! X are maps with ri D id (a retraction). If P is bijective for bundles over Y and is a bundle over X, we pull back this bundle to D r. From the naturality of P and the fact that P is bijective, we conclude that P is bijective. (iii) Let C be a compact smooth manifold with boundary. Let B denote the double D.C / of C. Then C is a retract of B. Hence P is bijective for bundles over C. 534 Chapter 21. Bordism (iv) Let X be a ﬁnite CW-complex.
natural isomorphism T.X/ W Nn.X/ Š M On.X C/: We will see that the isomorphism T.X/ is obtained by a stable version of the Pontrjagin–Thom construction. For each space X we denote by m.X/ the product bundle idX m. We deﬁne a map …k.X/ W L.RnCk; k.X//! Nn.X/: Let ŒM; F be an n-dimensional k.X/-submanifold of RnCk. The ﬁrst component F1 of the k.X/-structure F D.F1; F2/ W E.M /! X E./ gives us the element …k.X/ŒM; F D ŒM; F1 2 Nn.X/. It is obvious that we obtain a well-deﬁned map …k. There is a kind of suspension map W L.RnCk; k.X//! L.RnCkC1; kC1.X//: For ŒM; F consider M 0 D M f0g R nCk f0g R nCkC1. The normal bundle of M 0 is E.M / ˚ ". We compose E.M / ˚ "! E.k/ ˚ " with the classifying map E.k/ ˚ "! E.kC1/. From F we thus obtain a new structure F 0 D.F1; F 0 2/ W E.M 0/! X E.kC1/. We set ŒM; F D ŒM 0; F 0. The commutativity …kC1 D …k holds. Let Ln.X/ denote the colimit over the maps W L.RnCk; k.X//! L.RnCkC1; kC1.X//. Altogether we obtain ….X/ W Ln.X/! Nn.X/. (21.3.2) Proposition. The map ….X/ W Ln.X/! Nn.X/ is bijective. Proof. Surjective. Let ŒM; f 2 Nn.X/ be given. We can assume M RnCk for
some k, by the Whitney embedding theorem. Let M W M! k be a classifying map of the normal bundle. Then we have the k.X/-structure F D.f ı M ; M /, and …k.X/ŒM; F D ŒM; f holds by construction. 536 Chapter 21. Bordism Injective. Suppose ŒM0; F0 and ŒM1; F1 have the same image under ….X/. We can assume that Mj RnCk for a suitable k. There exists a bordism B with @B D M0 C M1 and an extension f W B! X of hf0; f1 i where fj is the ﬁrst component of Fj. There exists an embedding B RnCkCt Œ0; 1 such that B \.RnCk Rt Œ0; 1=3Œ / D M0 0 Œ0; 1=3Œ ; B \.RnCk Rt 2=3; 1/ D M1 0 2=3; 1: By use of collars we can ﬁnd a bordism B such that'W C D M0 Œ0; 1=2Œ C 1=2; 1 B and @B D M00CM11. We embed C! RnCk Rt Œ0; 1,.m; s/ 7!.m; 0; s/. Then we choose a continuous function ˆ W B! RnCkCt Œ0; 1 such that ˆ extends'on D D M0 Œ0; C M1 Œ1 ; 1 for some 1=3 < < 1=2 and such that ˆ.B X D/ is contained in RnCkCt 1=3; 2=3Œ (Tietze extension theorem). Suppose k C t > n C 1. We now approximate ˆ by an embedding J W B! RnCkCt Œ0; 1 such that J.B X D/ RnCkCt 1=3; 2=3Œ and such that J equals'on M0 Œ0; 1=3Œ CM1 2=3; 1. The bundle maps Mj! k yield bundle maps Mj ˚ t "!
omorphism n.X/ Š M SOn.X C/ is established as in the case of unoriented bordism. The Pontrjagin–Thom construction uses in this case an orientation of the normal bundle. An embedding M n RnCk induces a canonical isomorphism.M / ˚.M / Š.n C k/" in which the normal bundle.M / is the orthogonal complement of the tangent bundle. An orientation of.M / induces an orientation of.M / such that ﬁbrewise x.M / ˚ x.M / Š RnCk is orientation preserving. (21.4.1) Lemma. M SO.k/ is.k 1/-connected. Proof. The canonical map s W BSO.k 1/! BSO.k/ can be taken as the sphere bundle of the universal oriented k-dimensional bundle. From the homotopy sequence of this ﬁbration we see that s is.k 1/-connected. The homotopy sequence of s is isomorphic to the sequence of the pair.Dk; Sk/ of the universal (disk,sphere)-bundle over BSO.k/. Hence.Dk; Sk/ is.k 1/-connected. Since BSO.k/ is simply connected, we can apply (6.10.2) and see that j.Dk; Sk/! j.Dk=Sk/ D j.M SO.k// is an isomorphism for j k 1. The suspension isomorphism nCk.M SO.k//! nCkC1.†M SO.k// is an isomorphism for k n C 2, since M SO.k/ is.k 1/-connected (see (6.10.4)). The spectral map j.†M SO.k//! j.M SO.k C 1// is an isomorphism for j 2k 1. In order to see this, we use the Whitehead theorem (20.1.4): The spaces in question are simply connected. Thus it sufﬁces to see that we have a homology isomorphism in the same range. Hj.†M SO.k// Hj.M SO.k C 1// Thom Thom Hj k.BSO.k// Hj k.BSO.k C
1// The map BSO.k/! BSO.k C 1/ is.k 1/-connected, hence the vertical maps are isomorphisms for j k k 1. These arguments show that we need 538 Chapter 21. Bordism not pass to the colimit, colimk nCkM SO.k/; we already have an isomorphism n Š nCkM SO.k/ for k n C 2. (The geometric reason for this stability result is the strong form of the Whitney embedding theorem which we had not used in (21.3.1).) The assignment ŒM 7!.M /ŒM 2 Hn.BSOI Q/ of Section 19.8 induces a ring homomorphism! H.BSOI Q/ which we extend to a homomorphism of Q-algebras W ˝ Q! H.BSOI Q/. We can deﬁne in a similar manner a homomorphism W ˝ Q! H.BSOI Q/ if we use classifying maps of stable normal bundles. (21.4.2) Theorem. The homomorphisms W ˝ Q! H.BSOI Q/ and W ˝ Q! H.BSOI Q/ are isomorphisms of graded algebras. Proof. Lemma (21.4.1) and (20.8.3) imply that the Hurewicz homomorphism r.M SO.k//! Hk.M SO.k// has for r < 2k 1 both a ﬁnite kernel and a ﬁnite cokernel; hence it induces an isomorphism r.M SO.k// ˝ Q! Hr.M SO.k/I Q/ in this range. We also have the homological Thom isomorphism Hr.M SO.k/I Q/ Š Hrk.BSO.k/I Q/. The previous considerations now show that we have an isomorphism n ˝ Q Š Hn.BSO.k/I Q/. The computation of Hn.BSO.k/I Q/ Š Hn.BSOI Q/ for k n C 2 shows that the Q-vector space 4n ˝ Q has dimension.n/, the number of partitions of n.
From our computation of.CP 2a/ we see that W 4n ˝ Q! H4n.BSOI Q/ is a surjective map between vector spaces of the same dimension, hence an isomorphism. The homomorphism is obtained from by composition with the antipode of the Hopf algebra H.BSOI Q/. It is determined by the formal relation.1 C q1 C q2 C /.1 C.q1/ C.q2/ C / D 1 which expresses the relation between the Pontrjagin classes of a bundle and its inverse. Together with our previous computations (19.8.4) we obtain: (21.4.3) Theorem. The algebra ˝ Q is a polynomial Q-algebra in the generators ŒCP 2n, n 2 N. We now collect various results and prove the Hirzebruch signature theorem. (21.4.4) Theorem. The signature.M / of an oriented closed 4n-manifold is obtained by evaluating the Hirzebruch polynomial Ln in the Pontrjagin classes of M on the fundamental class. 21.4. Oriented Bordism 539 Proof. From (18.7.3) and (18.7.7) we see that M 7!.M / induces a ring homomorphism! Z. We extend it to a homomorphism of Q-algebras W! Q. Via the isomorphism of Theorem (21.4.2) it corresponds to a homomorphism s W H.BSOI Q/! Q such that D s ı. The homomorphism s was used at the end of Section 19.8 to determine polynomials Ln 2 H 4n.BSOI Q/ in the Pontrjagin classes such that hLn.p/; ŒM i D.M /. Problems 1. It is not necessary to use the computation of.CP 2a/ in the proof of (21.4.2). The reader is asked to check that the diagram P n nCk.M SO.k// h HnCk.M SO.k// Hn.BSO/ Š tZ Hn.BSO.k// commutes (at least up to sign if one does not care about speciﬁc orientations). P

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