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symbols X 0 D.X X U ı/ [h Dk S nk1: The transition from X to X 0 is called elementary surgery of index k at X via h. The method of surgery is very useful for the construction of manifolds with prescribed topological properties. See [191], [162], [108] to get an impression of Þ surgery theory. 15.10. Gluing along Boundaries 391 Problems 1. The subsets of S mCnC1 RmC1 RnC1 D1 D f.x; y/ j kxk2 1 2 ; kyk2 1 2 g; D2 D f.x; y/ j kxk2 1 2 ; kyk2 1 2 g p are diffeomorphic to D1 Š S m DnC1, D2 Š DmC1 S n. They are smooth submanifolds with boundary of S mCnC1. Hence S mCnC1 can be obtained from S m DnC1 and DmC1 S n by identifying the common boundary S m S n with the identity. A diffeomorphism D1! S m DnC1 is.z; w/ 7!.kzk1z; 2. Let M be a manifold with non-empty boundary. Identify two copies along the boundary with the identity. The result is the double D.M / of M. Show that D.M / for a compact M is the boundary of some compact manifold. (Hint: Rotate M about @M about 180 degrees.) 3. Show M #S n Š M for each n-manifold M. 4. Study the classiﬁcation of closed connected surfaces. The orientable surfaces are S 2 and connected sums of tori T D S 1 S 1. The non-orientable ones are connected sums of projective planes P D RP 2. The relation T #P D P #P #P holds. The connected sum with T is classically also called attaching of a handle. 2w/. Chapter 16 Homology of Manifolds The singular homology groups of a cell complex vanish above its dimension. It It is certainly is an obvious question whether the same holds for a manifold. technically complicated to produce a cell decomposition of a manifold and also an artiﬁcial structure. Locally the manifold looks like a Euclidean space
, so there arises no problem locally. The Mayer–Vietoris sequences can be used to paste local information, and we use this technique to prove the vanishing theorem. The homology groups of an n-manifold M in dimension n also have special properties. They can be used to deﬁne and construct homological orientations of a manifold. A local orientation about x 2 M is a generator of the local homology group Hn.M; M X xI Z/ Š Z. In the case of a surface, the two generators correspond to “clockwise” and “counter-clockwise”. If you pick a local orientation, then you can transport it along paths, and this deﬁnes a functor from the fundamental groupoid and hence a twofold covering. If the covering is trivial, then the manifold is called orientable, and otherwise (as in the case of a Möbius-band) non-orientable. Our ﬁrst aim in this chapter will be to construct the orientation covering and use it to deﬁne orientations as compatible families of local orientations. In the case of a closed compact connected manifold we can deﬁne a global homological orientation to be a generator of Hn.M I Z/; we show that this group is either zero or Z. In the setting of a triangulation of a manifold, the generator is the sum of the n-dimensional simplices, oriented in a coherent manner. In a non-orientable manifold it is impossible to orient the simplices coherently; but in that case their sum still gives a generator in Hn.M I Z=2/, since Z=2-coefﬁcients mean that we can ignore orientations. Once we have global orientations, we can deﬁne the degree of a map between oriented manifolds. This is analogous to the case of spheres already studied. 16.1 Local Homology Groups Let h./ be a homology theory and M an n-dimensional manifold. Groups of the type hk.M; M X x/ are called local homology groups. Let'W U! Rn be a chart of M centered at x. We excise M X U and obtain an isomorphism hk.M; M X x/ Š hk.U; U X x/ '! hk.Rn; Rn X 0
/: For singular homology with coefﬁcients in G we see that Hn.M; M X xI G/ Š G, and the other local homology groups are zero. Let R be a commutative ring. Then 16.1. Local Homology Groups 393 Hn.M; M X xI R/ Š R is a free R-module of rank 1. A generator, corresponding to a unit of R, is called a local R-orientation of M about x. We assemble the totality of local homology groups into a covering. Let K L M. The homomorphism r L K W hk.M; M XL/! hk.M; M XK/, x in the case that K D fxg. (16.1.1) Lemma. Each neighbourhood W of x contains an open neighbourhood U of x such that the restriction r U induced by the inclusion, is called restriction. We write r L y is for each y 2 U an isomorphism. Proof. Choose a chart'W V! Rn with V W centered at x. Set U D '1.En/, En D fx 2 Rn j kxk < 1g. We have a commutative diagram hk.M; M X U / r U y hk.M; M X y/.1/.2/ hk.V; V X U /.3/ hk.V; V X y/; with morphisms induced by inclusion. The maps (1) and (2) are excisions, and (3) is an isomorphism, because V X U V X y is for each y 2 U an h-equivalence (see Problem 1). We construct a covering! W hk.M; M X /! M. As a set hk.M; M X / D x2M hk.M; M X x/; ` and hk.M; M X x/ is the ﬁbre of! over x (with discrete topology). Let U be an open neighbourhood of x such that r U y is an isomorphism for each y 2 U. We deﬁne bundle charts 'x;U W U hk.M; M X x/!!1.U /;.y; a/ 7! r U y.r U x /1.a/
: We give hk.M; M X / the topology which makes 'x;U a homeomorphism onto an open subset. We have to show that the transition maps '1 y;V 'x;U W.U \ V / hk.M; M X x/!.U \ V / hk.M; M X y/ are continuous. Given z 2 U \ V, choose z 2 W U \ V such that r W isomorphism for each w 2 W. Consider now the diagram w is an hk.M; M X x/ r U x hk.M; M X U / r U w hk.M; M X w hk.M; M X W / r V W hk.M; M X V / r V y hk.M; M X y/: 394 Chapter 16. Homology of Manifolds y;V 'x;U D r V It shows '1 x /1. Hence the second component of '1 y;V 'x;U is independent of w 2 W, and this shows the continuity of the transition map. W /1r U W.r U y.r V We take advantage of the fact that the ﬁbres are groups. For A M we denote by.A/ the set of continuous (D locally constant) sections over A of! W hk.M; M X /! M. If s and t are sections, we can deﬁne.s C t/.a/ D s.a/ C t.a/. One uses the bundle charts to see that s C t is again continuous. Hence.A/ is an abelian group. We denote by c.A/.A/ the subgroup of sections with compact support, i.e., of sections which have values zero away from a compact set. (16.1.2) Proposition. Let z 2 hk.M; M X U /. Then y 7! r U section of!. y z is a continuous Proof. The bundle chart 'x;U transforms the constant section y 7!.y; r U the section y 7! r U x z/ into y z. Problems 1. S n1 Rn X E n and Rn X e! S n1, y 7!.y e/=ky ek are h-equivalences (e 2 E n).
The map S n1! S n1, y 7!.y e/=ky ek is homotopic to the identity. These facts imply that Rn X E Rn X e is an h-equivalence. 2. Let M be an n-dimensional manifold with boundary @M. Show that x 2 @M if and only if Hn.M; M Xx/ D 0. From this homological characterization of boundary points one obtains: Let f W M! M be a homeomorphism. Then f.@M / D @M and f.M X @M / D M X @M. 3. Let'W.C; 0/!.D; 0/ be a homeomorphism between open neighbourhoods of 0 in Rn. Then'W Hn.C; C X 0I R/! Hn.D; D X 0I R/ is multiplication by ˙1. 16.2 Homological Orientations Let M be an n-manifold and A M. An R-orientation of M along A is a section s 2.AI R/ of! W Hn.M; M X I R/! M such that s.a/ 2 Hn.M; M X aI R/ Š R is for each a 2 A a generator of this group. Thus s combines the local orientations in a continuous manner. In the case that A D M, we talk about an R-orientation of M, and for R D Z we just talk about orientations. If an orientation exists, we call M (homologically) orientable. If M is orientable along A and B A, then M is orientable along B. (16.2.1) Note. Let Ori.M / Hn.M; M X I Z/ be the subset of all generators of all ﬁbres. Then the restriction Ori.M /! M of! is a 2-fold covering of M, called the orientation covering of M. (16.2.2) Proposition. The following are equivalent: 16.2. Homological Orientations 395 (1) M is orientable. (2) M is orientable along compact subsets. (3) The orientation covering is trivial. (4) The covering! W Hn.M; M X I Z/! M is trivial. Proof. (1) ) (2). Special case.
(2) ) (3). The orientation covering is trivial if and only if the covering over each component is trivial. Therefore let M be connected. Then a 2-fold covering QM! M is trivial if and only if QM are also coverings. QM is not connected, hence the components of Suppose Ori.M /! M is non-trivial. Since Ori.M / is then connected, there exists a path in Ori.M / between the two points of a given ﬁbre. The image S of such a path is compact and connected, and the covering is non-trivial over S, since we can connect two points of a ﬁbre in it. By the assumption (2), the orientation covering is trivial over the compact set S, hence it has a section over S. Contradiction. (3) ) (4). Let s be a section of the orientation covering. Then M Z! Hn.M; M X I Z/,.x; k/ 7! ks.x/ is a trivialization of!: It is a map overM, bijective on ﬁbres, continuous, and a morphism between coverings. (4) ) (1). If! is trivial, then it has a section with values in the set of generators. (16.2.3) Note. Ori.M /! M is a twofold principal covering with automorphism group C D f1; t j t 2 D 1g. Let t act on G as multiplication by 1. Then the associated covering Ori(M) C G is isomorphic to Hn.M; M X I G/. Proof. The map Ori.M / G! Hn.M; M X I Z/ ˝ G Š Hn.M; M X I G/;.u; g/ 7! u ˝ g induces the isomorphism. (The isomorphism is the ﬁbrewise isomorphism Hn.M; M X xI Z/ ˝ G Š Hn.M; M X xI G/ from the universal coefﬁcient for- mula.) (16.2.4) Remark. The sections.AI G/ of! over A correspond bijectively to the continuous maps W Ori.M /jA! G with the property ı t D. This is a �
� general fact about sections of associated bundles. Problems 1. Let M be a smooth n-manifold with an orienting atlas. Then there exists a unique homological Z-orientation such that the local orientations in Hn.M; M X xI Z/ are mapped via positive charts to a standard generator of Hn.Rn; Rn X 0I Z/. Conversely, if M is Zoriented, then M has an orienting atlas which produces the given Z-orientation. 2. Every manifold has a unique Z=2-orientation. 396 Chapter 16. Homology of Manifolds 16.3 Homology in the Dimension of the Manifold Let M be an n-manifold and A M a closed subset. We use in this section singular homology with coefﬁcients in the abelian group G and sometimes suppress G in the notation. (16.3.1) Proposition. For each ˛ 2 Hn.M; M X AI G/ the section J A.˛/ W A! Hn.M; M X I G/; x 7! r A x.˛/ of! (over A) is continuous and has compact support. Proof. Let the chain c 2 Sn.M I G/ represent the homology class ˛. There exists a compact set K such that c is a chain in K. Let x 2 A X K. Then the image of c under Sn.KI G/! Sn.M I G/! Sn.M; KI G/! Sn.M; M X xI G/ is zero. Since this image represents r A A \ K. The continuity is a general fact (16.1.2). From (16.3.1) we obtain a homomorphism x.˛/, the support of J A.˛/ is contained in J A W Hn.M; M X AI G/! c.AI G/; ˛ 7!.x 7! r A x.˛//: (16.3.2) Theorem. Let A M be closed. (1) Then Hi.M; M X A/ D 0 for i > n. (2) The homomorphism J A W Hn.M; M X A/! c.A/ is an isomorphism. Proof. Let D.A; 1/
and D.A; 2/ stand for the statement that (1) and (2) holds for the subset A, respectively. We use the fact that J A is a natural transformation between contravariant functors on the category of closed subsets of M and their inclusions. The proof is a kind of induction over the complexity of A. It will be divided into several steps. (1) D.A; j /; D.B; j /; D.A \ B; j / imply D.A [ B; j /. For the proof we use the relative Mayer–Vietoris sequence for.M X A \ BI M X A; M X B/ and an analogous sequence for sections. This leads us to consider the diagram HnC1.M; M X.A \ B// Hn.M; M X.A [ B// Š J A[B 0 c.A [ B/ Hn.M; M X A/ ˚ Hn.M; M X B/ J A˚J B Š c.A/ ˚ c.B/ Hn.M; M X.A \ B// J A\B Š c.A \ B/: 16.3. Homology in the Dimension of the Manifold 397 The Five Lemma and the hypotheses now show D.A [ B; 2/. The Mayer–Vietoris sequence alone yields D.A [ B; 1/. (2) D.A; j / holds for compact convex subsets A in a chart domain U, i.e., '.A/ D B is compact convex for a suitable chart'W U! Rn. For the proof we show that for x 2 A the restriction r A x is an isomorphism. By an appropriate choice of'we can assume 0 2 B En. Then we have a commutative diagram Hi.Dn; S n1/.1/ Hi.Rn; Rn X B/ D Hi.Dn; S n1/.2/ Hi.Rn; Rn X 0/'Š'Š Hi.U; U X A/.3/ Hi.M; M X A/ Hi.U; U X x/.4/ r A x Hi.M; M X x/. The maps (3) and (4) are excisions. The maps (1)
and (2) are isomorphisms, because S n1 Rn X B is an h-equivalence. The isomorphism r A x shows, ﬁrstly, that D.A; 1/ holds; and, secondly, D.A; 2/, since a section of a covering over a connected set is determined by a single value. (3) Suppose A U,'W U! Rn a chart, A D K1 [ [ Kr, '.Ki / compact convex. We show D.A; j / by induction on r. Let B D K1 [ [ Kr1 and C D Kr. Then B and B \ C are unions of r 1 sets of type (2), hence D.B; j / and D.B \ C; j / holds by induction. Now use (1). (4) Let K U,'W U! Rn a chart, K compact. Let K W U, W open. Then there exists a neighbourhood V of K inside W of type (3). In this case J V is an isomorphism. The restrictions r V K induce canonical maps from the colimits colimV Hi.M; M X V / colimV J V colimV c.V /././ Hi.M; M X K/ J K c.K/, where the colimit is taken over the directed set of neighbourhoods V of K of type (3). What does this isomorphism statement mean in explicit terms? Firstly, an element xK in the image has the form r V K xV for a suitable V ; and secondly, if xV and xW have the same image xK, then they become equal under a restriction to a suitable smaller neighbourhood. From this description it is then easy to verify that J K is indeed an isomorphism. Suppose./ and./ are isomorphisms. Then we obtain D.K; 1/, and the isomorphisms J V yield D.K; 2/. The isomorphism./ holds already for the singular chain groups. It uses the fact that a chain has compact support; if the support is contained in M X K, then already in M X V for a suitable neighbourhood V of K../ is an isomorphism: See Problem 1. (5) D.K; j / holds for arbitrary compact subsets K, for K is a union of a ﬁnite number of sets of type
(4). Then we can use induction as in case (3). 398 Chapter 16. Homology of Manifolds (6) Let K D K1 [ K2 [ K3 [ with compact Ki. Suppose there are pairwise disjoint open neighbourhoods Ui of Ki. By additivity of homology groups and section groups, J K is the direct sum of the J Ki. (7) Let A ﬁnally be an arbitrary closed subset. Since M is locally compact with countable basis, there exists an exhaustion M D [Ki, K1 K2 by compact sets Ki such that Ki Kı i1/, K0 D ;, B D iD2nC1 Ai. Then D.B; j /; D.C; j /, and D.B \ C; j / hold by (6); the hypothesis of (6) follows from the fact that a manifold is a normal space. Now D.A; j / holds by (1), since A D B [ C. iC1. Set Ai D A \.Ki X Kı iD2n Ai, C D S S (16.3.3) Theorem. Suppose A is a closed connected subset of M. Then: (1) Hn.M; M X AI G/ D 0, ifA is not compact. (2) Hn.M; M X AI G/ Š G, if M is R-orientable along A and A is compact. Moreover Hn.M; M X AI G/! Hn.M; M X xI G/ is an isomorphism for each x 2 A. (3) Hn.M; M X AI Z/ Š 2G D fg 2 G j 2g D 0g, if M is not orientable along A and A is compact. Proof. (1) Since A is connected, a section in.AI G/ is determined by its value at a single point. If this value is non-zero, then the section is non-zero everywhere. Therefore there do not exist non-zero sections with compact support over a noncompact A, and (16.3.2) shows Hn.M; M X AI G/ D 0. (2) Let A be compact. Then Hn.M; M X AI G/ Š.AI G/. Again a section is determined by its value at a single point
. We have a commutative diagram Hn.M; M X AI G/ Š.AI G/ r A x Hn.M; M X xI G/ Š b.fxgI G/. If M is orientable along A, then there exists in.A/ an element such that its value at x is a generator. Hence b is an isomorphism and therefore also r A x. (3) A section in.AI G/ corresponds to a continuous map W Ori.M /jA! G with t D. If M is not orientable along A, then Ori.M /jA is connected and therefore constant. The relation t D shows that the value of is contained in 2G. In this case r A x W Hn.M; M X AI G/! Hn.M; M X xI G/ Š G is injective and has image 2G. Theorem (16.3.3) can be considered as a duality result, since it relates an asser- tion about A with an assertion about M X A. (16.3.4) Proposition. Let M be an n-manifold and A M a closed connected subset. Then the torsion subgroup of Hn1.M; M X AI Z/ is of order 2 if A is compact and M non-orientable along A, and is zero otherwise. 16.4. Fundamental Class and Degree 399 Proof. Let q 2 N and suppose M is orientable along the compact set A. Then Z=q Š Hn.M; M X AI Z=q/ Š Hn.M; M X AI Z/ ˝ Z=q ˚ Hn1.M; M X AI Z/ Z=q Š Z=q ˚ Hn1.M; M X AI Z/ Z=q: We have used: (16.3.3); universal coefﬁcient theorem; again (16.3.3). This implies that Hn1.M; M X AI Z/ Z=q D 0. Similarly for non-compact A or q odd 0 Š Hn.M; M X AI Z=q/ Š Hn1.M; M X AI Z/ Z=q: Since Tor.G; Z=q/ Š fg 2 G j qg D 0g
, this shows that Hn1.M; M X AI Z/ has no q-torsion in these cases. If A is compact and M non-orientable along A, then Z=2 Š Hn.M; M X AI Z=4/ Š Hn1.M; M X AI Z/ Z=4 by (16.3.3) and the universal coefﬁcient formula. Since we know already that the group in question has no odd torsion, we conclude that there exists a single non-zero element of ﬁnite order and the order is 2. Problems 1. Let s be a section over the compact set K. For each x 2 K there exists an open neighbourhood U.x/ and an extension sx of sjU.x/ \ K. Cover K by U.x1/; : : : ; U.xr /. Let W D fy j sxi.y/ D sxj.x/ if y 2 U.xi / \ U.xj /g and deﬁne s.y/; y 2 W as the common value. Show that W is open. Let s; s0 be sections over V which agree on K. Show that they agree in a smaller neighbourhood V1 V of K. (These assertions hold for sections of coverings.) 16.4 Fundamental Class and Degree The next theorem is a special case of (16.3.3). (16.4.1) Theorem. Let M be a compact connected n-manifold. Then one of the following assertions holds: (1) M is orientable, Hn.M / Š Z, and for each x 2 M the restriction Hn.M /! Hn.M; M X x/ is an isomorphism. (2) M is non-orientable and Hn.M / D 0. Under the hypothesis of (16.4.1), the orientations of M correspond to the generators of Hn.M /. A generator will be called fundamental class or homological orientation of the orientable manifold. We now use fundamental classes in order to deﬁne the degree; we proceed as in the special case of S n. Let M and N be compact oriented n-manifolds. Let 400 Chapter 16. Homology of Manifolds N be connected and suppose M has components M1;
: : : ; Mr. Then we have j Hn.Mj / is fundamental classes z.Mj / for the Mj and z.M / 2 Hn.M / Š the sum of the z.Mj /. For a continuous map f W M! N we deﬁne its degree d.f / 2 Z via the relation fz.M / D d.f /z.N /. From this deﬁnition we see immediately the following properties of the degree: L (1) The degree is a homotopy invariant. (2) d.f ı g/ D d.f /d.g/. (3) A homotopy equivalence has degree ˙1. (4) If M D M1 C M2, then d.f / D d.f jM1/ C d.f jM2/. (5) If we pass in M or N to the opposite orientation, then the degree changes the sign. We now come to the computation of the degree in terms of local data of the map. Let M and N be connected and set K D f 1.p/. Let U be an open neighbourhood of K in M. In the commutative diagram z.M / 2 Hn.M / f Hn.N / 3 z.N / Hn.M; M X K/ Š z.U; K/ 2 Hn.U; U X K/ f Š Hn.N; N X p/ D f U Hn.N; N X p/ 3 z.N; p/ we have f U z.U; K/ D d.f /z.N; p/. Thus the degree only depends on the restriction f U of f to U. One can now extend the earlier investigations of self maps of S n to this more general case. The additivity of the degree is proved in exactly the x2K Ux where the Ux are pair-wise same manner. Let K be ﬁnite. Choose U D disjoint open neighbourhoods of x. We then have S L x2K Hn.Ux; Ux X x/ Š Hn.U; U X K/; Hn.Ux; Ux X x/ Š Z: The image z.Ux; x/ of z.M / is a generator, the
local orientation determined by the fundamental class z.M /. The local degree d.f; x/ of f about x is deﬁned by fz.Ux; x/ D d.f; x/z.N; p/. The additivity yields d.f / D x2K d.f; x/. P (16.4.2) Remark. Let f be a C 1-map in a neighbourhood of x; this shall mean the following. There exist charts'W Ux! Rn centered at x and W V! Rn centered at p such that f.Ux/ V and g D f '1 is a C 1-map. We can suppose that the charts preserve the local orientations; this shall mean for'that'W Hn.Ux; Ux X x/! Hn.Rn; Rn X 0/ sends z.Ux; x/ to the standard generator. Such charts are called positive with respect to the given orientations. Suppose now in addition that the differential of g at x is regular. Then d.f; x/ is the sign of the Þ determinant of the Differential Dg.0/. 16.4. Fundamental Class and Degree 401 (16.4.3) Proposition. Let M be a connected, oriented, closed n-manifold. Then there exists for each k 2 Z a map f W M n! S n of degree k. Proof. If f W M! S n has degree a and g W S n! S n degree b, then gf has degree ab. Thus it sufﬁces to realize a degree ˙1. Let'W Dn! M be an embedding. Then we have a map f W M! Dn=S n1 which is the inverse of'on U D '.E n/ and sends M X U to the base point. This map has degree ˙1. Let the manifolds M and N be oriented by the fundamental classes zM 2 Hm.M / and zN 2 Hn.N /. Then the homology product zM zN is a fundamental class for M N, called the product orientation. Problems 1. Let p W M! N be a covering of n-manifolds. Then the pullback of Ori.N /! N along p is Ori.M /! M. 2. Let p W
M! N be a G-principal covering between connected n manifolds with orientable M. Then N is orientable if and only if G acts by orientation-preserving homeomorphisms. 3. The manifold Ori.M / is always orientable. 4. RP n is orientable if and only if n is odd. 5. Let M be a closed oriented connected n-manifold. Suppose that M carries a CWdecomposition with k-skeleton Mk. The inclusion induces an injective map Hn.M /! Hn.Mn; Mn1/. The fundamental class is therefore represented by a cellular chain in Hn.Mn; Mn1/. If we orient the n-cells in accordance with the local orientations of the manifold, then the fundamental class chain is the sum of the n-cells. This is the classical interpretation of the fundamental class of a triangulated manifold. A similar assertion holds for unoriented manifolds and coefﬁcients in Z=2 and manifolds with boundary (to be considered in the next section). In a sense, a similar assertion should hold for non-compact manifolds; but the cellular chain would have to be an inﬁnite sum. Therefore a fundamental class has to be deﬁned via an inverse limit. 6. Let M be a closed connected n-manifold. Then Hn.M I Z=2/ Š Z=2 and the restrictions x W Hn.M I Z=2/! Hn.M; M X xI Z=2/ are isomorphisms. r M 7. Let f W M! N be a map between closed connected n-manifold. Then one can deﬁne the degree modulo 2 d2.f / 2 Z=2; it is zero (one) if f W Hn.M I Z=2/! Hn.N I Z=2/ is the zero map (an isomorphism). If this degree is non-zero, then f is surjective. If the manifolds are oriented, then d.f / mod 2 D d2.f /. 8. Let G be a compact connected Lie group and let T be a maximal torus of G. The map q W G=T T! G,.g; t/ 7! gtg1 has degree jW j. Here W D N T =T
is the Weyl group. Since q has non-zero degree, this map is surjective (see [29, IV.1]). 9. Let G be a compact connected Lie group and T a maximal torus of G. The degree of f W G! G, g 7! gk has degree kr, r D dim T. Let c 2 T be an element such that the powers of c are dense in T, then jf 1.c/j Dk r, f 1.c/ T, and c is a regular value of f. [90] 10. Let f W M! N be a proper map between oriented connected n-manifolds. Deﬁne the degree of f. 402 Chapter 16. Homology of Manifolds 16.5 Manifolds with Boundary Let M be an n-dimensional manifold with boundary. We call z 2 Hn.M; @M / a fundamental class if for each x 2 M X @M the restriction of z is a generator in Hn.M; M X x/. (16.5.1) Theorem. Let M be a compact connected n-manifold with non-empty boundary. Then one of the following assertions hold: (1) Hn.M; @M / Š Z, and a generator of this group is a fundamental class. The image of a fundamental class under @ W Hn.M; @M /! Hn1.@M / is a fundamental class. The interior M X @M is orientable. (2) Hn.M; @M / D 0, and M X @M is not orientable. Proof. Let W Œ0; 1Œ @M! U be a collar of M, i.e., a homeomorphism onto an open neighbourhood U of @M such that.0; x/ D x for x 2 @M. For simplicity of notation we identify U with Œ0; 1Œ @M via ; similarly for subsets of U. In this sense @M D 0 @M. We have isomorphisms Hn.M; @M / Š Hn.M; Œ0; 1Œ @M / Š Hn.M X @M; 0; 1Œ@M / Š.A/: The ﬁrst one by h-equivalence; the second one by excision;
the third one uses the closed set A D M X.Œ0; 1Œ @M / M X @M and (16.3.2). The set A is connected, hence.A/ Š Z or.A/ Š 0. If.A/ Š Z, then M X@M is orientable along A. Instead of A we can argue with the complement of Œ0; "Œ @M. Since each compact subset of M X @M is contained in some such complement, we see that M X @M is orientable along compact subsets, hence orientable (see (16.2.2)). The isomorphism Hn.M X @M; 0; 1Œ @M / Š.A/ says that there exists an element z 2 Hn.M X @M; 0; 1Œ @M / which restricts to a generator of Hn.M X @M; M X @M X x/ for each x 2 A. For the corresponding element z 2 Hn.M; @M / a similar assertion holds for each x 2 M X @M, i.e., z is a fundamental class (move around x within the collar). It remains to show that @z is a fundamental class. The lower part of the diagram (for x 2 0; 1Œ @M ) Hn1.@M / Š Hn1.@M [ A; A/ Š Hn1.@I @M; 1 @M / @ @ Š @ Š Hn.M; @M / Hn.M; @M [ A/ Š Hn.I @M; @I @M / ))))))))))))) Hn.M; M X x/ Š Hn.I @M; I @M X x/ 16.6. Winding and Linking Numbers 403 shows that z yields a fundamental class in Hn.I @M; @I @M /. The upper part shows that this fundamental class corresponds to a fundamental class in Hn1.@M /, since fundamental classes are characterized by the fact that they are generators (for each component of @M ). (16.5.2) Example. Suppose the n-manifold M is the boundary of the compact orientable.nC1/-manifold B. We have the fundamental classes z.B/ 2 HnC
1.B; @B/ and z.M / D @zB 2 Hn.M /. Let f W M! N be a map which has an extension F W B! N, then the degree of f (if deﬁned) is zero, d.f / D 0, for we have fz.M / D f@z.B/ D Fi@z.B/ D 0, since i@ D 0 as consecutive morphisms in the exact homology sequence of the pair.B; M /. We call maps f W M! N orientable bordant if there exists a compact oriented manifold B with oriented boundary @B D M1 M2 and an extension F W B! N of hf1; f2 i W M1 C M2! N. The minus sign in @B D M1 M2 means @z.B/ D z.M1/ z.M2/. Under these assumptions we have d.f1/ D d.f2/. This fact is called the bordism invariance Þ of the degree; it generalizes the homotopy invariance. Problems 1. Let M be a compact connected n-manifold with boundary. Then Hn.M; @M I Z=2/ is isomorphic to Z=2; the non-zero element is a Z=2-fundamental class z.M I Z=2/. The restriction to Hn.M; M XxI Z=2/ is for each x 2 M X@M an isomorphism and @z.M I Z=2/ is a Z=2-fundamental class for @M. 2. Show that the degree d2.f / is a bordism invariant. 16.6 Winding and Linking Numbers Let M be a closed connected oriented n-manifold. Let f W M! RnC1 and a … Im.f /. The winding number W.f; a/ of f with respect to a is the degree of the map pf;a D pa W M! S n; x 7! N.f.x/ a/ where N W RnC1 X 0! S n, x 7! kxk1x. If ft is a homotopy with a … Im.ft / for each t, then W.f0; a/ D W.ft ; a/. (16
.6.1) Theorem. Let M be the oriented boundary of the compact smooth oriented manifold B. Let F W B! RnC1 be smooth with regular value 0 and assume 0 … f.M /. Then W.f; 0/ D P x2P ".F; x/; P D F 1.0/; f D F j@B where ".F; x/ 2 f˙1g is the orientation behaviour of the differential TxF W TB! T0.RnC1/. 404 Chapter 16. Homology of Manifolds P P S Proof. Let D.x/ BX@B, x 2 P be small disjoint disks about x. Then G D N ıF is deﬁned on C D B x2P D.x/, and by the bordism invariance of the degree d.Gj@B/ D x2P d.Gj@D.x//. By (16.4.2), d.Gj@D.x// D ".F; x/. Let M and N be oriented closed submanifolds of RkC1 of dimensions m and n with k D m C n. Let M N carry the product orientation. The degree of the map fM;N D f W M N! S k;.x; y/ 7! N.x y/ is the linking number L.M; N / of the pair.M; N /. More generally, if the maps W M m! RkC1 and W N n! RkC1 have disjoint images, then the degree of.x; y/ 7! N..x/.y// is the linking number of.; /. Problems 1. Let the n-manifold M be the oriented boundary of the smooth connected compact manifold B. Suppose f W M! S n has degree zero. Then f can be extended to B. 2. Show L.M; N / D.1/mCnC1L.N; M /. 3. Let f; g W R! R3 be smooth embeddings with disjoint closed images. Deﬁne a linking number for the pair.f; g/ and justify the deﬁnition. Chapter 17 Cohomology The axioms for a cohomology theory are analogous to the axioms of a homology theory
. Now we consider contravariant functors. The reader should compare the two deﬁnitions, also with respect to notation. One advantage of cohomology is an additional internal product structure (called cup product) which will be explained in subsequent sections. The product structure suggests to view the family.hn.X/ j n 2 Z/ as a single object; the product then furnishes it with the structure of a ring (graded algebra). Apart from the additional information in the product structure, the ring structure is also notationally convenient (for instance, a polynomial ring has a better description than its additive group without using the multiplicative structure). Singular cohomology is obtained from the singular chain complex by an application of the Hom-functor. We present an explicit deﬁnition of the cup product in singular cohomology (Alexander–Whitney). In more abstract terms the product can also be obtained from the Eilenberg–Zilber chain equivalences as in the case of the homology product. We use the product structure to prove a powerful theorem (Leray–Hirsch) which says roughly that the cohomology of the total space of a ﬁbration is a free module over the cohomology ring of the base, provided the ﬁbre is a free module and a basis of the ﬁbre-cohomology can be lifted to the total space. In the case of a topological product, the result is a special case of the Künneth theorem if the cohomology of the ﬁbre is free, since the Ext-groups vanish in that case. One interesting application is to vector bundles; the resulting so-called Thom isomorphism can be considered as a twisted suspension isomorphism in that the suspension is replaced by a sphere bundle. As a speciﬁc example we determine the cohomology rings of the projective spaces. 17.1 Axiomatic Cohomology 17.1.1 The axioms. A cohomology theory for pairs of spaces with values in the category of R-modules consists of a family.hn j n 2 Z/ of contravariant functors hn W TOP(2)! R- MOD and a family.ın j n 2 Z/ of natural transformations ın W hn1 ı! hn. These data are required to satisfy the following axioms. (1
) Homotopy invariance. Homotopic maps f0 and f1 between pairs of spaces induce the same homomorphism, hn.f0/ D hn.f1/. 406 Chapter 17. Cohomology (2) Exact sequence. For each pair.X; A/ the sequence! hn1.A; ;/ ı! hn.X; A/! hn.X; ;/! hn.A; ;/ ı! is exact. The undecorated arrows are induced by the inclusions. (3) Excision. Let.X; A/ be a pair and U A such that xU Aı. Then the inclusion.X X U; A X U /!.X; A/ induces an excision isomorphism hn.X; A/ Š hn.X X U; A X U /. We call hn.X; A/ the n-th cohomology group of.X; A/. The ın are called the coboundary operators. We write hn.X; ;/ D hn.X/ and hn.f / D f. Occasionally we refer to the homomorphisms i W hn.X/! hn.A/ induced by an inclusion i W A X as restriction. The groups hn.P / Š hn for a point P are said to be the coefﬁcient groups of the theory (compatible family of isomorphisms to a given module hn). In the case that hn.P / D 0 for n 6D 0, we talk about an ordinary or classical cohomology theory and say that the theory satisﬁes the dimension axiom. The notation hn.X; A/ D H n.X; AI G/ stands for an ordinary cohomology theory with a given isomorphism h0.P / Š G. The cohomology theory is additive if ` j Xj ; ` j Aj! hn Q j hn.Xj ; Aj /; x 7!.hn.ij /.x// is always an isomorphism (ij the inclusion of the j -th summand). For ﬁnite J the Þ additivity isomorphism follows from the other axioms. Several formal consequences of the homology axioms have analogues in coho- mology and
the proofs are similar. We mention some of them. We begin with the exact sequence of a triple.X; A; B/. The coboundary operator is in this case deﬁned by ı W hn1.A; B/! hn1.A/! hn.X; A/. The ﬁrst map is induced by the inclusion and the second map is the given coboundary operator. For each triple.X; A; B/ the sequence! hn1.A; B/ ı! hn.X; A/! hn.X; B/! hn.A; B/ ı! is exact. The undecorated arrows are restrictions. 17.1.2 Suspension. The suspension isomorphism is deﬁned by the commutative diagram hn.Y; B/ hnC1..I; @I /.Y; B// Š ı Š hn.0 Y; 0 B/ Š hn.@I Y [ I B; 1 Y [ I B/: For homology we used a deﬁnition with the roles of 0; 1 interchanged. Þ 17.1. Axiomatic Cohomology 407 17.1.3 Reduced cohomology. The reduced cohomology groups of a non-empty space X are deﬁned as Qhn.X/ D coker.p W hn.P /! hn.X// where p W X! P denotes the unique map to a point. The functors Qhn./ are homotopy invariant. For a pointed space.X; / we have the canonical split exact sequence 0! hn.X; / j! hn.X/ i! hn./! 0: The restriction j induces an isomorphism hn.X; / Š Qhn.X/ and we have isomorphisms hj; p i W hn.X; / ˚ hn Š hn.X/;.q; i/ W hn.X/ Š Qhn.X/ ˚ hn with the quotient map q. The coboundary operator ın W hn1.A/! h.X; A/ factors over the quotient map q W hn1.A/! Qhn1.A
/. Passing to quotients yields the exact sequence! Qhn1.A/ ı! h.X; A/! Qhn.X/! Qhn.A/ ı! for the reduced groups. Þ 17.1.4 Mayer–Vietoris sequence. A triad.XI A; B/ is excisive for the cohomology theory if the inclusion induces an isomorphism h.A [ B; A/ Š h.B; A \ B/. This property can be characterized in different ways as in the case of a homology theory, see (10.7.1) and (10.7.5). In particular the property is symmetric in A; B. We have exact Mayer–Vietoris sequences for excisive triads. As in the case of a homology theory one can derive some MV-sequences by diagram chasing. For the general case of two excisive triads we use a method which we developed in the case of homology theory; the MV-sequence was obtained as the exact sequence of a triad of auxiliary spaces by some rewriting (see (10.7.6)). This procedure also works for cohomology. Let.AI A0; A1/.XI X0; X1/ be excisive triads. Set X01 D X0 \ X1 and A01 D A0 \ A1. Then there exists an exact Mayer–Vietoris sequence of the following form hn.X01; A01/.1/ hn.X0; A0/ ˚ hn.X1; A1/.2/ hn.X; A/ hn1.X01; A01/ : The map.1/ is.x0; x1/ 7! i 1 x1 with the inclusions i; the components of.2/ are the restrictions. The connecting morphism in the case A D A0 D A1 is the composition 0 x0 i W hn1.X01; A/ ı! hn.X0; X01/ Š hn.X; X1/! hn.X; A/ 408 Chapter 17. Cohomology and the connecting morphism in the case X D X0 D X1 is W hn.X; A01/! hn.A0; A01/ Š hn.A; A1/ ı
! hnC1.X; A/: Þ 17.1.5 Limits. We have seen that an additive homology theory is compatible with colimits. The situation for cohomology and limits is more complicated. Let G W G1 G2 momorphisms. The group p1 Q p2 G3 p3 : : : i2N Gi acts on the set i2N Gi be a sequence of groups and ho- Q.g1; g2; : : : /.h1; h2; : : : / D.g1h1p1.g2/1; g2h2p2.g3/1; : : : /: The orbit set is denoted lim1.G/ D lim1.Gi ; pi / D lim1.Gi /. A direct consequence of this deﬁnition is: Suppose the groups Gi are abelian. Then we have an exact sequence 0! lim.Gi ; pi /! Q i2N Gi Q d! i2N Gi! lim1.Gi ; pi /! 0: Here d.g1; g2; : : : / D.g1 p1.g2/; g2 p2.g3/; : : : / and lim is the limit of the sequence. As a consequence of the ker-coker-sequence we obtain: A short exact sequence 0!.G0 i ; p0 i /!.Gi ; pi /!.G00 i ; p00 i /! 0 of inverse systems induces an exact sequence 0! lim0.G0! lim1.G0 i /! lim0.Gi /! lim0.G00 i / i /! lim1.Gi /! lim1.G00 i /! 0: Let now an additive cohomology theory by given. We apply the Mayer–Vietoris sequence to the telescope T of a sequence X1 X2 of spaces with colimit X. The result is [133]: (17.1.6) Proposition. There exists an exact sequence 0! lim1.hn1.Xi //! hn.T /! lim.hn.Xi //! 0: Proof. We use the MV-sequence of the triad.T I A; B/ as in (10.8.2). It has the form! hn.T /! hn.A/
˚ hn.B/ and yields the short exact sequence ˇ n! hn.A \ B/! 0! Coker.ˇn1/! hn.T /! Ker.ˇn/! 0: 17.2. Multiplicative Cohomology Theories 409 Thus we have to determine the kernel and the cokernel. By additivity of the cohomology theory we obtain Y Y Y hn.A/ Š hn.Xi /; hn.B/ Š hn.Xi /; hn.A \ B/ Š hn.Xi /: i0.2/ i1.2/ i2N These isomorphisms transform ˇn into.x1; x2; x3; : : : / 7!.x1 C f 1.x2/; x2 f 2.x3/; : : : /: The isomorphism.ai / 7!..1/i ai / transforms this map ﬁnally into the map d in the deﬁnition of lim and lim1. (17.1.7) Proposition. Suppose the homomorphisms pi between the abelian groups Gi are surjective. Then lim1.Gi ; pi / D 0. i2N Gi. We have to show that g is contained Proof. Let g D.g1; g2; : : : / 2 in the image of d, i.e., we have to solve the equations gi D xi pi.xiC1/ with suitable xi 2 Gi. This is done inductively. Q Problems 1. The system.Gi ; pi / satisﬁes the Mittag-Lefﬂer condition (D ML) if for each i there exists j such that for k j the equality Im.GiCk! Gi / D Im.GiCj! Gi / holds. If.Gi ; pi / satisﬁes ML, then lim1.Gi ; pi / D 0. Thus if the groups Gi are ﬁnite, then lim1.Gi ; pi / D 0. If the Gi are countable and if lim1.Gi ; pi / D 0, then the system satisﬁes ML ([44, p. 154]). 2. Imitate the earlier investigation of cellular homology and show that H.
X/ can be determined from a cellular cochain complex which arises from a cellular decomposition of X. 3. Let W k./! l./ we a natural transformation between additive cohomology theories which induces isomorphisms of the coefﬁcient groups. Show that is an isomorphism for each CW-complex. 17.2 Multiplicative Cohomology Theories Let h be a cohomology theory with values in R- MOD. A multiplicative structure on this theory consists of a family of R-linear maps (m; n 2 Z) hm.X; A/ ˝R hn.X; B/! hmCn.X; A [ B/; x ˝ y 7! x Y y; deﬁned for suitable triads.XI A; B/, and in any case for excisive.A; B/ in X. We call x Y y the cup product of x; y. The products are always deﬁned if A or B is 410 Chapter 17. Cohomology empty or if A D B. In this section, tensor products will be taken over R. The cup product maps are required to satisfy the following axioms. (1) Naturality. For maps of triads f W.XI A; B/!.X 0I A0; B 0/ the commutativity f.x Y y/ D f x Y f y holds. (2) Stability. Let.A; B/ be excisive in X. We use the restriction morphism A W hj.X; B/! hj.A; A \ B/ and the coboundary operator ıA W hr.A; A \ B/ Š hr.A [ B; B/! hrC1.XI A [ B/. The diagram ı hi.A/ ˝ hj.X; B/ 1˝A hi.A/ ˝ hj.A; A \ B/ Y hiCj.A; A \ B/ ı˝1 ıA hiC1.X; A/ ˝ hj.X; B/ Y hiCj C1.X; A [ B/ is commutative. (3) Stability. Let.A; B/ be excisive in X. We use the restriction morphism B W hi.X;
A/! hi.B; A \ B/ and the coboundary operator ıB W hr.B; A \ B/ Š hr.A [ B; A/! hrC1.X; A [ B/. The diagram ı hi.X; A/ ˝ hj.B/ B ˝1 hi.B; A \ B/ ˝ hj.B/ Y hiCj.B; A \ B/ 1˝ı ıB hi.X; A/ ˝ hj C1.X; B/ Y hiCj C1.X; A [ B/ is commutative up to the sign.1/i. (4) Unit element. There is given a unit 1 2 h0.P /, P a ﬁxed point, as additional It induces 1 D 1X D p.1/ 2 h0.X/, p W X! P. Then structure datum for each x 2 hm.X; A/. (5) Associativity..x Y y/ Y z D x Y.y Y z/. (6) Commutativity. x Y y D.1/jxjjyjy Y x. One can also consider situations where (6) or (5) do not hold. This is the reason for requiring (2) and (3) separately. For (5) it is required that the products are deﬁned. For the convenience of the reader we also display the properties in a table and refer to the detailed description above. 17.2. Multiplicative Cohomology Theories 411 f.x Y y/ D f x Y f y ı.a/ Y x D ıA.a Y Ax/ x Y ı.b/ D.1/jxjıB.B x Y bx Y y/ Y z D x Y.y Y z/ x Y y D.1/jxjjyjy Y x The cup product deﬁnes on h.X; A/ the structure of a Z-graded associative and If A D ; this algebra has a unit element 1X 2 h0.X/. commutative algebra. Moreover h.X; A/ becomes a unital graded left h.X/-module. These structures have the obvious naturality properties which follow from the axioms, e
.g., a map f W X! Y induces a unital algebra homomorphism f W h.Y /! h.X/. In particular the graded module h D.hn/ of the coefﬁcient groups becomes a unital commutative graded algebra. We make h.X; A/ into a unital left h-module via a x D p.a/ Y x with p W X! P the unique map to a point. Morphisms induced by continuous maps are then h-linear. This is a particular case of the cross product introduced later. We list some consequences of naturality and stability. Let y 2 hn.X; B/ be ﬁxed. Right multiplication by y yields a morphism of degree n from the exact sequence of the pair into the exact sequence of the triple.X; A [ B; B/. This means: We have a commutative diagram hm.X/ ry hmCn.X; B/ hm.A/ ry hmCn.A [ B; B/ ı ı hmC1.X; A/ ry hmCnC1.X; A [ B/: The ry in the middle is deﬁned by multiplication of x 2 hm.A/ with the restriction of y along hn.X; B/! hn.A; A \ B/ and then using hn.A; A \ B/ Š hn.A [ B; B/. Let y 2 hn.X/ be ﬁxed. For each pair.A; B/ in X we obtain a product ry W hk.A; B/! hkCn.A; B/ by right multiplication with the restriction of y along hn.X/! hn.A/. The following diagram commutes: hm.A/ ry hmCn.A/ hm.B/ ry hmCn.B/ ı ı hmC1.A; B/ ry hmCnC1.A; B/. If.A; B/ is excisive, then the coboundary operators of the MV-sequences W hm1.A \ B; C /! hm.A; A \ B/ Š hm.A [ B; B/! hm.
A [ B; C /; 412 Chapter 17. Cohomology W hm.X; A \ B/! hm.A; A \ B/ Š hm.A [ B; B/! hmC1.X; A \ B/ commute with ry. Also in the general case 17.1.4, the boundary operator commutes with products: (17.2.1) Proposition. Let.AI A0; A1/.XI X0; X1/ be excisive triads. Let X 0 X and assume that also.A[X 0I A0 [X 0 D X \X 0. Then the diagram 1/ is excisive for X 0 0 ; A1 [A0 hr.X01; A01/ hrC1.X; A/ ry ry hrCjyj.X01; A01 [ X 0 01/ hrC1Cjyj.X; A [ X 0/ commutes. Here ry is right multiplication by y 2 h.X; X 0/ (bottom) and multiplication with the restriction of y to h.X01; X 0 01/ (top). Proof. The coboundary operators are deﬁned via suspension and appropriate induced morphisms. The commutativity of the diagram then amounts to the naturality of the cup product and a compatibility (17.2.2) with the suspension. (17.2.2) Proposition. The diagram h.I Y; IB [ @I Y / ˝ h.I Y; I Y 0/ Y h.I Y; I.B [ I 0/ [ @I Y / ˝pr h.Y; B/ ˝ h.Y; Y 0/ Y h.Y; B [ Y 0/ commutes. (Notation: I Y D I Y; @I Y D @I Y etc.) Proof. We use the associated cross product and (17.3.1), (17.3.3) in the computation.e1 x/ Y.1I y/ D.e1 Y 1I /.x Y y/. (17.2.3) Example. Let.X; / be a pointed space. The ring homomorphism i W h.X/! h has as kernel the two-sided h-ideal h.X; /. We have the isomorph
ism h.X; / ˚ h Š h.X/,.a; b/ 7! ja C pb, see 17.1.3. This isomorphism transforms the cup product on h.X/ into the product.a1; b1/.a2; b2/ D.a1 Y a2 C b1a2 C a1b2; b1b2/: Þ (17.2.4) Example. The commutativity x Y x D.1/jxjjxjx Y x has for jxj 1 mod 2 the consequence 2.x Y x/ D 0. Hence x Y x D 0, if multiplication by 2 Þ is injective. 17.3. External Products 413 (17.2.5) Example. Let X D A1 [ [ Ak. Suppose ai 2 hn.i/.X/ are elements in the kernel of the restriction hn.i/.X/! hn.i/.Ai /. Let bi 2 hn.i/.X; Ai / be a pre-image of ai. Then a1 Y Y ak D 0, because this element is the image of b1 Y Y bk 2 hn.X; i Ai / D 0. This argument requires that the relative products are deﬁned. S This simple consequence of the existence of products has interesting geometric consequences. The projective space CP n has a covering by n C 1 afﬁne (hence contractible) open sets Uj D fŒzi j zj 6D 0g. Therefore a0 Y Y an D 0 for elements aj 2 hn.j /.CP n; /. Later we show the existence of an element c 2 H 2.CP nI Z/ with cn 6D 0. Hence CP n cannot be covered by n contractible open sets (more generally, by open sets U such that U! CP n is null homotopic). An analogous result holds for RP n. Þ (17.2.6) Example. The argument of the preceding example shows that products in h.S n; /, n 1 are trivial, a1 Y a2 D 0. More generally, let X be a well-pointed space. Then products in h.†X; / are trivial. It sufﬁces to prove this fact for the unreduced
suspension †0X; but this space has a covering by two contractible open sets (cones). Additively, the cohomology groups only depend on the stable homotopy type. Þ The product structure contains more subtle information. (17.2.7) Example. Let p W E! B be any map. Then we make h.E/ into a graded right h.B/- module by the deﬁnition y x D y Y px for x 2 h.B/ and y 2 h.E/. If f W X! Y is a morphism from p W X! B to q W Y! B in TOPB, then f W h.Y /! h.X/ is h.B/-linear. The same device works for pairs of spaces over B. The coboundary operator is then also h.B/-linear, Þ ı.y x/ D ıy x. 17.3 External Products A multiplicative structure on a cohomology theory h of external products consists of a family of R-linear maps (m; n 2 Z) hm.X; A/ ˝R hn.Y; B/! hmCn..X; A/.Y; B//; x ˝ y 7! x y; deﬁned for a suitable class of pairs.X; A/ and.Y; B/ and in any case if the pair.X B; A Y / is excisive in X Y. The products are deﬁned if A or B is empty. These maps are required to satisfy the following axioms. (1) Naturality. For continuous maps f W.X; A/!.X 0; A0/ and g W.Y; B/!.Y 0; B 0/ we have.f g/.x y/ D f x gy. (2) Stability. Let.X B; A Y / be excisive. For x 2 hm.A/ and y 2 hn.Y; B/ the relation ıx y D ı0.x y/ holds. Here ı0 is the composition of the excision 414 Chapter 17. Cohomology isomorphism with ı for k D m C n ı0 W hk.A.Y; B// Š hk.A Y [ X B; X B/
ı! hkC1..X; A/.Y; B//: (3) Stability. Let.X B; A Y / be excisive. For x 2 hm.X; A/ and y 2 hn.B/ the relation x ıy D.1/jxjı00.x y/ holds. Here ı00 is a similar composition of excision with ı for k D m C n ı00 W hk..X; A/ B/ Š hk.X B [ A Y; A Y / ı! H kC1..X; A/.Y; B//: (4) Unit element. There is given 1 2 h0.P / as a further structure datum. It satisﬁes 1 x D x 1 D x (with respect to P X D X P D X). (5) Associativity..x y/ z D x.y z/. (6) Commutativity. Let W X Y! Y X,.x; y/ 7!.y; x/. Then for x 2 hm.X; A/ and y 2 hn.Y; B/ the relation.x y/ D.1/jxjjyjy x holds. Also in this case we display the properties in a table and refer to the detailed explanation above..f g/.x y/ D f x gy ıx y D ı0.x y/ x ıy D.1/jxjı00.x y/ 1 x D x D x 1.x y/ z D x.y z/.x y/ D.1/jxjjyjy x The term “stability” usually refers to compatibility with suspension. We explain this for the present setup. As a ﬁrst application we show that the suspension isomorphism is given by multiplication with a standard element. Let e1 2 h1.I; @I / be the image of 1 2 h0 under 1 2 h0 Š h0.0/ Š h0.@I; 1/ ı h1.I; @I / 3 e1: (17.3.1) Proposition. e1 y D.y/. Proof. Consider the diagram hm.Y; B/ Q1 hm.@I Y; @IB/ hm.
0Y; 0B/ hmC1..I; @I /.Y; B/ ı hm.@I Y [ IB; IB/ hm.@I Y [ IB; 1Y [ IB/: ˛ Š ˇ Š 17.3. External Products 415 Let Q1 2 h0.@I / be the image of 1 2 h0.0/ under h0.0/ Š h0.@I; 1/! h0.@I /. Then e1 D ı.Q1/. Stability (2) of the cross product shows e1 y D ı˛1.Q1 y/. The maps ˛ and ˇ are isomorphisms by excision and h-equivalence. The outer diagram path is y 7!.y/. Suppose we have a cup product. We construct an associated external product. Let p W.X; A/ Y!.X; A/ and q W X.Y; B/!.Y; B/ be the projections onto the factors. We deﬁne for x 2 hm.X; A/ and y 2 hn.Y; B/ the product x y D py [ qy. Conversely, suppose an external product is given. We deﬁne an associated Y-product. Let d W.X; A [ B/!.X; A/.X; B/ be the diagonal. Then we set x Y y D d.x y/. (17.3.2) Proposition. The -product associated to a Y-product satisﬁes the axioms of an external product. The Y-product associated to a -product satisﬁes the axioms of an internal product. The processes Y and Y are inverse to each other. (17.3.3) Proposition. Let xi 2 h.X; Ai / and yi 2 h.Y; Bi /. Then.x1 y1/ Y.x2 y2/ D.1/jx2jjy1j.x1 Y x2/.y1 Y y2/: In particular h.X/ ˝ h.Y /! h.X Y /, x ˝ y 7! x y is a homomorphism of unital graded algebras. (17.3.4)
Proposition. Let s 2 hn.S n; / hn.S n/ be the element which corresponds to 1 2 h0 under a suspension isomorphism. Then for each space F the map hk.F / ˚ hkn.F /! hk.F S n/;.a; b/ 7! a 1 C b s is an isomorphism. These isomorphisms show that h.F S n/ is a free graded left h.F /-module with homogeneous basis 1 1, 1 s. Proof. We start with the isomorphism hkn.F /! hk.I n F; @I n F /; x 7! en x where en D e1 e1, see (17.3.1). Let s 2 hn.S n; / correspond to en under the isomorphism hn.S n; / Š hn.I n=@I n; / Š hn.I n; @I n/. Then, by naturality, hkn.F /! hk.S n F; F /; b 7! s b is an isomorphism. We now use the split exact sequence 0! hk.S n F; F /! hk.S n F /! hk.F /! 0 416 Chapter 17. Cohomology with splitting hk.F /! hk.S n F /, a 7! 1 a D pr.a/ in order to see that.a; b/ 7! 1 a C s b is an isomorphism. Under this isomorphism we have the following expression of the product struc- ture since s2 D 0..a C sb/.a0 C sb0/ D aa0 C s.ba0 C.1/jajnab0/ We use commutativity to show that.a; b/ 7! a 1 C b s is an isomorphism as claimed. If we use 1 D 1F 1S n and s D 1F s as basis elements for the left h.F /-module h.F S n/, then the h.F / algebra is seen to be the graded exterior algebra h.F /Œs=.s2/. Let B be a CW-complex. The skeleton ﬁltration.F khi.B/ j k 2 N/ on
hi.B/ is deﬁned by F khi.B/ D Ker.hi.B/! hi.B k1//. (17.3.5) Proposition. The skeleton ﬁltration is multiplicative: If a 2 F khi.B/ and b 2 F l hj.B/, then a Y b 2 F kCl hiCj.B/. Proof. Choose pre-images a0 2 hi.B; B k1/, b0 2 hl.B; B l1/. Then a0 b0 2 hiCj.B B; B k1 B [ B B l1/. The product a Y b is the image of a b under the diagonal d. A cellular approximation d 0 W B! B B of the diagonal d sends B kCl1 into.B B/kCl1 and the latter is contained in B k1 B [ B B l1. Naturality of the -product now shows that a Y b is contained in the image of hiCj.B; B kCl1/! hiCj.B/. (17.3.6) Corollary. Let n 2 h0.B/ be contained in F 1h0.B/. Then nk 2 F kh0.B/. Thus if B is ﬁnite-dimensional, n is nilpotent and 1 C n a unit. Problems 1. Supply the proofs for (17.3.2) and (17.3.3). 2. Determine the algebras h.S n.1/ S n.k// as graded h-algebras (graded exterior algebra). 17.4 Singular Cohomology The singular cohomology theory is constructed from the singular chain complexes by a purely algebraic process. The algebraic dual of the singular chain complex is again a chain complex, and its homology groups are called cohomology groups. It is customary to use a “co” terminology in this context. Let C D.Cn; @n/ be a chain complex of R-modules. Let G be another Rmodule. We apply the functor HomR.; G/ to C and obtain a chain complex C D.C n; ın/ of R-modules with C n D HomR.Cn; G/ and the R-linear map ın W C n D HomR.Cn;
G/! HomR.CnC1; G/ D C nC1 17.4. Singular Cohomology 417 deﬁned by ın.'/ D.1/nC1' ı @nC1 for'2 Hom.Cn; G/. For the choice of this sign see 11.7.4. The reader will ﬁnd different choices of signs in the literature. Other choices will not effect the cohomology functors. But there seems to be an agreement that our choice is the best one when it comes to products. For a pair of spaces.X; A/ we have the singular chain complex S.X; A/. With an abelian group G we set S n.X; AI G/ D Hom.Sn.X; A/; G/ and use the coboundary operator above. Here we are using HomZ. Elements in S n.X; AI G/ are functions which associate to each singular simplex W n! X an element of G and the value 0 to simplices with image in A. If G is an R-module, then the set of these functions becomes an R-module by pointwise addition and scalar multiplication. A continuous map f W.X; A/!.Y; B/ induces homomorphisms f # D S n.f / W S n.Y; BI G/! S n.X; AI G/;.f #'/./ D '.f / which are compatible with the coboundary operators. In this manner we obtain a contravariant functor from TOP.2/ into the category of cochain complexes of R-modules. The n-th cohomology group of S.X; AI G/ is denoted H n.X; AI G/ and called the n-th singular cohomology module of.X; A/ with coefﬁcients in G. We often write H n.X; AI Z/ D H n.X; A/ and talk about integral cohomology, in this case (G D Z=.p/ mod-p cohomology). Dualization of the split exact sequence 0! S.A/! S.X/! S.X; A/! 0 yields again an exact sequence 0! S.X; AI G/! S.XI G/! S.AI G/! 0. It induces
a long exact cohomology sequence! H n1.AI G/ ı! H n.X; AI G/! H n.XI G/! H n.AI G/! : (17.4.1) Remark. We recall the deﬁnition of the coboundary operator ı for the present situation. Let'W Sn1.A/! G be a cocycle, i.e.,'ı @ D 0. Extend'in an arbitrary manner to a function Q' W Sn1.X/! G. The element ı. Q'/ D.1/n Q' ı @ W Sn.X/! G vanishes on Sn.A/, since its restriction to Sn.A/ is'ı @. Therefore ı. Q'/ yields an element 2 S n.X; AI G/. The coboundary operator is Þ then deﬁned by the assignment Œ' 7! Œ. So far we have deﬁned the data of a cohomology theory. If we apply Hom.; G/ to a chain homotopy we obtain a cochain homotopy; this yields the homotopy invariance. The excision axiom holds, as the chain equivalence S.X XU; AXU /'S.X; A/ induces a chain equivalence S.X; AI G/'S.X X U; A X U I G/. There exist several algebraic relations among homology and cohomology. The ﬁrst one comes from the evaluation of the Hom-complex. Let us use coefﬁcients in a commutative ring R. The evaluations Hom.Sn.X; A/; R/ ˝ Sn.X; AI R/! R;'˝.c ˝ r/ 7! '.c/r 418 Chapter 17. Cohomology induces a pairing (sometimes called Kronecker pairing) H n.X; AI R/ ˝ Hn.X; AI R/! R; x ˝ y 7! hx; y i: This is due to the fact that the evaluations combine to a chain map, see 11.7.4. (17.4.2) Proposition. For f W.X; A/!.Y; B/, y 2 H n.Y; BI R/, and
x 2 Hn.X; AI R/ we have hf.y/; x i D hy; f.x/i: For a 2 H n1.AI R/, b 2 Hn.X; AI R/ we have hıa; b i C.1/ n1ha; @b i D0: Proof. We verify the second relation. Let a D Œ','2 Hom.Sn1.A/; Z/. Let Q' W Sn1.X/! Z be an extension of '. Then ı.a/ is represented by the homomorphism.1/n Q'@ W Sn.X/! Z (see (17.4.1)). Let y D Œc, c 2 Sn.X/. Then hıx; y i D.1/ n Q'@.c/. From @.c/ 2 Sn1.A/ we conclude Q'.@.c// D '.@c/ D hx; @y i. The canonical generator e1 2 H1.I; @I / is represented by the singular simplex s W 1! I,.t0; t1/ 7! t1. Let Œx 2 H 0.X/ denote the element represented by the cochain which assumes the value 1 on x 2 X and 0 otherwise. (17.4.3) Proposition. Let e1 be the generator which is the image of Œ0 2 H 0.0/ under H 0.0/ H 0.@I; 1/ ı! H 1.I; @I /. Then he1; e1 i D1. The singular cohomology groups H n.X; AI G/ can be computed from the homology groups of.X; A/. This is done via the universal coefﬁcient formula. We have developed the relevant algebra in (11.9.2). The application to topology starts with the chain complex C D S.X; AI R/ D S.X; A/ ˝Z R of singular chains with coefﬁcients in a principal ideal domain R. It is a complex of free R-modules. Note that HomR.Sn.X; A/ ˝Z R; G/ Š HomZ.Sn.X; A/; G/ where in the second group G is considered as abelian group (D Z-
module). Then (11.9.2) yields the universal coefﬁcient formula for singular cohomology. (17.4.4) Theorem. For each pair of spaces.X; A/ and each R-module G there exists an exact sequence 0! Ext.Hn1.X; AI R/; G/! H n.X; AI G/! Hom.Hn.X; AI R/; G/! 0: The sequence is natural in.X; A/ and in G and splits. isomorphisms H 0.XI G/ Š Hom.H0.X/; G/ Š Map.0.X/; G/. In particular, we have The statement that the splitting is not natural means that, although the term in the middle is the direct sum of the adjacent terms, this is not a direct sum of functors. There is some additional information that cannot be obtained directly from the homology functors. The topological version of (11.9.6) is: 17.5. Eilenberg–Mac Lane Spaces and Cohomology 419 (17.4.5) Theorem. Let R be a principal ideal domain. Assume that either H.X; AI R/ is of ﬁnite type or the R-module G is ﬁnitely generated. Then there is a functorial exact sequence 0! H n.X; AI R/ ˝ G! H n.X; AI G/! Tor.H nC1.X; AI R/; G/! 0: The sequence splits. (17.4.6) Proposition. Let M be a closed, connected, non-orientable n-manifold. Then H n.M I Z/ Š Z=2. Proof. Since M is non-orientable, Hn.M I Z/ D 0. Theorem (17.4.4) for R D G D Z then shows H n.M I Z/ Š Ext.Hn1.M I Z/; Z/. From (16.3.4) we know that Hn1.M I Z/ Š Z=2 ˚ F with a ﬁnitely generated free abelian group. This is also a consequence of Poincaré duality, see (18.3.4). Problems 1. Let 0!
G1! G2! G3! 0 be an exact sequence of abelian groups. It induces a short exact sequence of cochain complexes 0! Hom.S.X/; G1/! Hom.S.X/; G2/! Hom.S.X/; G3/! 0 and an associated long exact sequence of cohomology groups. The coboundary operator ˇ W H n.XI G3/! H nC1.XI G1/ is a natural transformation of functors H n.I G3/! H nC1.I G1/ and called a Bockstein operator. A typical and interesting case arises from the exact sequence 0! Z=p! Z=p2! Z=p! 0. 2. If the functor Hom.; G/ preserves exact sequences, then no Ext-term appears in the universal coefﬁcient formula. Examples are H n.X; AI Q/ Š HomZ.Hn.X; A/; Q/ Š HomQ.Hn.X; AI Q/; Q/ and H n.X I Z=p/ Š HomZ=p.Hn.XI Z=p/; Z=p/ for the prime ﬁeld Z=p as coefﬁcient ring. 17.5 Eilenberg–Mac Lane Spaces and Cohomology The representability theorem (8.6.10) of Brown can be used to ﬁnd a natural isomorphism W Œ; K.A; n/! H n.I A/ on the homotopy category of CW-complexes. In this section we construct this isomorphism and give some applications. A natural transformation is determined by its value on the identity of K.A; n/, and this value can be prescribed arbitrarily. Thus we have to ﬁnd a suitable element n 2 H n.K.A; n/I A/. 420 Chapter 17. Cohomology Let us begin with the special case n D 0. As a model for K.A; 0/ we take the abelian group A with discrete topology. Since A is discrete, a map X! A is continuous if and only if it is locally constant. Moreover, all homotopies are constant. Therefore ŒX; K.A; 0/ D ŒX; A
is the group of locally constant maps. A locally constant map is constant on a path component. Therefore the surjection q W X! 0.X/ induces an injective homomorphism 0.X/ W ŒX; A! Map.0.X/; A/ Š Hom.H0.X/; A/ Š H 0.XI A/: The last isomorphism is the one that appears in the universal coefﬁcient formula. If X is locally path connected, then the homomorphism induced by q is an isomorphism. Hence we have obtained a natural isomorphism on the category of locally path connected spaces, in particular on the category of CW-complexes. If X is connected but not path connected, then 0.X/ is not an isomorphism. Let now n 1 and write K D Kn D K.A; n/. Consider the composition n 2 H n.KI A/.1/ Š Hom.Hn.KI Z/; A/.2/ Š Hom.n.A/; A/.3/ Š Hom.A; A/ 3 id : The isomorphism (1) is the universal coefﬁcient isomorphism. The isomorphism (2) is induced by the Hurewicz isomorphism h W n.K; /! Hn.KI Z/, see (20.1.1). It sends Œf 2 ŒS.n/; K0 D n.K/ to f.zn/ where zn 2 Hn.S.n/I Z/ is a suitable generator. The isomorphism (3) is induced by a ﬁxed polarization W A Š n.K/. We deﬁne n as the element which corresponds to the identity of A. Let n be the natural transformation which is determined by the condition nŒid D n. Note that category theory does not tell us yet that the n.X/ W ŒX; K! H n.XI A/ are homomorphisms of abelian groups. Let us compare n1 and n. It sufﬁces to consider connected CW-complexes and pointed homotopy classes. The diagram with the structure map e.n/ W †Kn1! Kn of the spectrum ŒX; Kn10 † Œ†X; �
�Kn10 e.n/ Œ†X; Kn0 n1 QH n1.XI A/ n QH n.†XI A/ is commutative up to sign, provided e.n/.n/ D n.e.n// D ˙.n1/. The morphism n.†X/ is a homomorphism, since the group structures are also induced by the cogroup structure of †X. In order to check the commutativity one has to arrange and prove several things: (1) The Hurewicz homomorphisms commute with suspensions. (2) The structure map e.n/ and the polarizations satisfy e.n/ ı † ı n1 D n W A! n.Kn/. (3) The homomorphisms from the universal coefﬁcient formula commute up to sign with suspension. Since the sign is not important for the moment, we do not go into details. 17.5. Eilenberg–Mac Lane Spaces and Cohomology 421 (17.5.1) Theorem. The transformation n is an isomorphism on the category of pointed CW-complexes. Proof. We work with connected pointed CW-complexes. Both functors have value 0 for S m; m 6D n. The reader is asked to trace through the deﬁnitions and verify that n.S n/ is an isomorphism. By additivity, n.X/ is an isomorphism for X a wedge of spheres. Now one uses the coﬁbre sequence W S k1! X k1! X k! X k=X k1! †X k1 W and applies the functors Œ; Kn0 and H n.I A/. We use induction on k. Then n S k1, X k1, and X k=X k1. The Five Lemma implies is an isomorphism for that n.X k/ and hence also n.†X k1/ are surjective. By another application of the Five Lemma we see that n.X k/ is also injective. This settles the case of ﬁnite-dimensional CW-complexex. The general case follows from the fact that both functors yield an isomorphism when applied to X nC1 X. A CW-complex K.Z; n/ can be obtained by
attaching cells of dimension n C 2 to S n. The cellular approximation theorem (8.5.4) tells us that the inclusion i n W S n K.Z; n/ induces for each CW-complex X of dimension at most n a bijection W ŒX; S n i n We combine this with the isomorphism and obtain a bijection Š! ŒX; K.Z; n/: ŒX; S n Š! H n.XI Z/: It sends a class Œf 2 ŒX; S n to the image of 1 2 H n.S nI Z/ under f W H n.S n/! H n.X/. Here we use the isomorphism A Š H n.S nI A/ which is (for an arbitrary abelian group A) deﬁned as the composition H n.S nI A/ Š Hom.Hn.S n/; A/ Š Hom.n.S n/; A/ Š Hom.Z; A/ Š A: Again we have used universal coefﬁcients and the Hurewicz isomorphism. (17.5.2) Proposition (Hopf). Let M D X be a closed connected n-manifold which has an n-dimensional CW-decomposition. Suppose M is oriented by a fundamental class zM. Then we have an isomorphism H n.M I Z/ Š Hom.Hn.M /; Z/ Š Z where the second isomorphism sends ˛ to ˛.zM /. The isomorphism ŒM; S n Š H n.M I Z/ Š Z maps the class Œf to the degree d.f / of f. (17.5.3) Proposition (Hopf). Let M be a closed, connected and non-orientable n-manifold with a CW-decomposition. Then H n.M I Z/ Š Z=2; hence we have a bijection ŒM; S n Š Z=2. It sends Œf to the degree d2.f / modulo 2 of f. 422 Chapter 17. Cohomology Proof. The universal coefﬁcient theorem and Hn.M I Z/ D 0 show Ext.Hn1.M
/; Z=2/ Š H n.M I Z=2/: We use again that Hn1.M I Z/ Š F ˚ Z=2 with a ﬁnitely generated free abelian group F. This proves H n.M I Z=2/ Š Z=2. Naturality of the universal coefﬁcient sequences is used to show that the canonical map H n.M I Z/! H n.M I Z=2/ is an isomorphism. The commutative diagram H n.S nI Z=2/ f H n.M I Z=2/ Hom.Hn.S nI Z=2/; Z=2/ Hom.Hn.M I Z=2/; Z=2/ is used to show the assertion about the degree. Problems 1. is not always an isomorphism: A D Z, n D 1 and the pseudo-circle. 17.6 The Cup Product in Singular Cohomology Let R be a commutative ring. The cup product in singular cohomology with coefﬁcients in R arises from a cup product on the cochain level S k.XI R/ ˝ S l.XI R/! S kCl.XI R/;'˝ 7!'Y : It is deﬁned by.1/.' Y /./ D.1/j'jj j'.jŒe0; : : : ; ek/.jŒek; : : : ; ekCl /: Here W kCl D Œe0; : : : ; ekCl! X is a singular.kCl/-simplex. Let Œv0; : : : ; vn be an afﬁne n-simplex and W Œv0; : : : ; vn! X a continuous map. We denote by jŒv0; : : : ; vn the singular simplex obtained from the composition of with the map n! Œv0; : : : ; vn, ej 7! vj. This explains the notation in (1). (17.6.1) Proposition. The cup product is a chain map S.XI R/ ˝ S.XI R/! S.XI R/; i
.e., the following relation holds: ı.' Y / D ı' Y C.1/j'j' Y ı : 17.6. The Cup Product in Singular Cohomology 423 Proof. From the deﬁnition we compute.ı' Y /./ to be.1/.j'jC1/j j kX iD0.1/iCj'jC1'.jŒe0; : : : ; bei; : : : ; ekC1/.jŒekC1; : : : ; ekClC1/ and.1/j'j.' Y ı /./ to be.1/j'j.j jC1/ kClC1X.1/iCj jC1'.jŒe0; : : : ; ek/.jŒek; : : : ; bei ; : : : ; ekClC1/: iDk If we add the two sums, the last term of the ﬁrst sum and the ﬁrst term of the second sum cancel, the remaining terms yield.1/j'jCj jC1.' Y /.@/, and this equals.ı.' Y //. /. We extend the cup product to the relative case. Suppose'2 S k.X; AI R/ and'2 S l.X; BI R/ are given. This means:'vanishes on singular simplices k! A and vanishes on simplices l! B. The relation (17.6.1) then shows that'[ vanishes on the submodule SkCl.A C BI R/ SkCl.XI R/ generated by simplices in A and B. The pair.A; B/ is excisive for singular homology if S.A C B/ D S.A/ C S.B/ S.A [ B/ is a chain equivalence. The dual maps S.A [ BI R/! S.A C BI R/ and S.X; A [ BI R/! S.XI A C BI R/ are then chain equivalences. The last module consists of the cochains which vanish on S.A C B/; let H.XI A C BI R/ denote the corresponding cohomology group. Thus we obtain a cup product (again a
chain map) S.X; AI R/ ˝ S.X; BI R/! S.XI A C BI R/: We pass to cohomology, obtain H.X; AI R/˝H.X; BI R/! H.XI ACBI R/ and in the case of an excisive pair a cup product H.X; AI R/ ˝ H.X; BI R/! H.X; A [ BI R/: We now prove that the cup product satisﬁes the axioms of Section 2. From the deﬁnition (1) we see that naturality and associativity hold on the cochain level. The unit element 1X 2 H 0.XI R/ is represented by the cochain which assumes the value 1 on each 0-simplex. Hence it acts as unit element on the cochain level. In the relative case the associativity holds for the representing cocycles in the group H.XI A C B C C I R/. In order that the products are deﬁned one needs that the pairs.A; B/,.B; C /,.A [ B; C / and.A; B [ C / are excisive. Commutativity does not hold on the cochain level. We use: The homomor- phisms W Sn.X/ 7! Sn.X/; 7! "nxn 424 Chapter 17. Cohomology with "n D.1/.nC1/n=2 and x D jŒen; en1; : : : ; e0 form a natural chain map which is naturally chain homotopic to the identity (see (9.3.5) and Problem 1 in that section). The cochain map # induced by satisﬁes #' Y # D.1/j'jj j#.' Y /: Since # induces the identity on cohomology, the commutativity relation follows. The stability relation (2) is a consequence of the commutativity of the next diagram (which exists without excisiveness). Coefﬁcients are in R. H i.A/˝H j.X; B/ H i.A/˝H j.A; A; A \ B/ Y H iCj.A; A \ B/ ı˝1
ı 0 H iC1.X; A/˝H j.X; B/ Y H iCj C1.XI A C B/ where ı0 is the composition of H iCj.A; A \ B/ induced by the algebraic isomorphism Š H iCj.A C BI B/, which is S.A/=S.A \ B/ Š S.A C B/=S.B/; and ı W H iCj.A C BI B/! H iCj C1.XI A C B/. In order to verify the commutativity, one has to recall the construction of ı, see (17.4.1). Suppose Œ' 2 H i.A/ and Œ 2 H j.X; A/ are given. Let Q' 2 S i.X/ be an extension of '; then ı Q' vanishes on Si.A/, and the cochain ı Q' 2 S iC1.X; A/ represents ıŒ'. The image of Œ' ˝ Œ along the down-right path is represented by ı Q' [, and one veriﬁes that the image along the right-down path is represented by ı. Q' [ /. Since is a cocycle, the representing elements coincide. A similar veriﬁcation can be carried out for stability (3). One can deal with products from the view-point of Eilenberg–Zilber transfor- mations. We have the tautological chain map S.XI R/ ˝ S.Y I R/! Hom.S ˝ S; R ˝ R/: We compose it with the ring multiplication R ˝ R! R and an Eilenberg–Zilber transformation S.X Y /! S.X/ ˝ S.Y / and obtain a chain map S.XI R/ ˝ S.Y I R/! S.X Y I R/; f ˝ g 7! f g; a -product on the cochain level. If the pair.A Y; X B/ is excisive, we obtain a -product S.X; AI R/ ˝ S.Y; BI R/! S..X; A/.Y;
B/I R/: Our previous explicit deﬁnition of the cup product arises in this manner from the Alexander–Whitney equivalence and the related approximation of the diagonal. We can now apply the algebraic Künneth theorem for cohomology to the singular cochain complexes and obtain: 17.7. Fibration over Spheres 425 (17.6.2) Theorem. Let R be a principal ideal domain and H.Y; BI R/ of ﬁnite type. Assume that the pair.A Y; X B/ is excisive. Then there exists a natural short exact sequence L 0!! L iCj Dn H i.X; AI R/ ˝ H j.Y; BI R/! H n..X; A/.Y; B/I R/ iCj DnC1 H i.X; AI R/ H j.Y; BI R/ and this sequence splits. Problems 1. We have deﬁned the cup product for simplicity by explicit formulas. One can use instead Eilenberg–Zilber morphisms. Let us consider the absolute case. Consider the composition S q.XI R/ ˝ S q.Y I R/ D Hom.SpX; R/ ˝ Hom.Sq.Y /; R/! Hom.Sp.X/ ˝ Sq.Y /; R/! Hom.SpCq.X Y /; R/ where the ﬁrst morphism is the tautological map and the second induced by an Eilenberg– Zilber morphism S.X Y /! S.X/ ˝ S.Y /. Then this composition induces the -product in cohomology. The cup product in the case X D Y is obtained by composition with S.X /! S.X X/ induced by the diagonal. Instead we can go directly from Hom.Sp.X / ˝ Sq.X/; R/ to Hom.SpCq.X/; R/ by an approximation of the diagonal. Our previous deﬁnition used the Alexander–Whitney diagonal. 17.7 Fibration over Spheres Let p W X! S n be a ﬁbration (n 2/. We write as usual S n D Dn and S n1 D Dn ˙/, X0
D p1.S n1/, and F D p1.b0/. From the homotopy theorem of ﬁbrations we obtain the following result.. Let b0 2 S n1 be a base point. We set X˙ D p1.Dn C \ Dn C [ Dn (17.7.1) Proposition. There exist h-equivalences '˙ W Dn ˙ X! X˙ over Dn ˙ such that '˙.b0; y/ D y for y 2 F. The h-inverses ˙ also satisfy ˙.y/ D.b0; y/ and the ﬁbrewise homotopies of ˙'˙ and '˙ ˙ to the identity are constant on F. Since X0 X˙ X are closed coﬁbrations, we have a Mayer–Vietoris sequence for.XC; X/. We use the data of (17.7.1) in order to rewrite the MV-sequence. We work with a ˙ F, y 7!.b0; y/ multiplicative cohomology theory. The embedding j W F! Dn is an h-equivalence. Therefore we have isomorphisms ˙ W hk.X˙/ i'˙ Š hk.Dn ˙ F / j Š hk.F /: 426 Chapter 17. Cohomology The restriction of 'C gives us another isomorphism C W hk.X0/'We insert these isomorphisms into the MV-sequence of.XC; X/ Š! hk.S n1 F /: hk.X/ hk.XC/ ˚ hk.X/ D i C˚i hk.X0/'C hk.X/.1/ hk.F / ˚ hk.F /.2/ hk.S n1 F /. The two components of (1) equal i where i W F X. The ﬁrst component of (2) is induced by the projection pr W S n1 F! F. We write ı 'C in the form.s; y/ 7!.s; ˛.s; y//. Then the second
component of (2) is ˛. Both maps yield the identity when composed with j W F! S n1 F, y 7!.b0; y/. The product structure provides us with an isomorphism hk.F / ˚ hknC1.F /! hk.S n1 F /;.a; b/ 7! 1 a C s b: We also use this isomorphism to change the MV-sequence. We set ˛.x/ D 1 x s ‚.x/; ‚.x/ 2 hknC1.F /: The relation j.1 a s b/ D a shows that ˛.x/ has the displayed form. (17.7.2) Theorem (Wang Sequence). There exists an exact sequence! hk.X/ i! hk.F / ‚! hknC1.F /! hknC1.X/! : The map ‚ is a derivation, i.e., ‚.x [ y/ D ‚.x/ [ y C.1/jxj.n1/x [ ‚y. Proof. We start with the modiﬁed MV-sequence! hk.X/.1/! hk.F / ˚ hk.F /.2/! hk.F / ˚ hknC1.F /! : The morphism (1) is as before, and (2) has the form.a; b/ 7!.a b; ‚.b//. Then we form the quotients with respect to the left hk.F / summands in order to obtain the stated exact sequence. Since ˛ is a homomorphism and s2 D 0, we obtain ˛.xy/ D 1 xy C s ‚.xy/ ˛.x/˛.y/ D.1 x C s ‚.x//.1 y C s ‚.y// D 1 xy C s ‚.x/ y C.1/jxjjsjs x ‚.y/: This proves the derivation property of ‚. We can, of course, also consider the MV-sequence in homology. It assumes after an analogous rewriting the form! Hq.F / i! Hq.X
/! Hqn.F / ‚! Hq1.F /! : Problems 17.8. The Theorem of Leray and Hirsch 427 1. As an example for the use of the Wang sequence compute the integral cohomology ring p! S nC1 H.S nC1/ of the loop space of S nC1. Use the path ﬁbration S nC1 with contractible P.! P i If n D 0, then S 1 is h-equivalent to the discrete space Z. So let n 1. Since P is contractible, the Wang sequence yields Hq.S nC1/ Š Hqn.S nC1/ and therefore Hk.S nC1/ Š Z for k 0 mod.n/ and Š 0 otherwise. Similarly for cohomology H k.S nC1/ Š Z for k 0 mod.n/ and zero otherwise. Using the isomorphism ‚ we deﬁne inductively elements z0 D 1 and ‚zk D zk1 for k 1. Let n be even. Then kŠzk D zk 1 for k 1. For the proof use induction over k and the derivation property of ‚. The relation above yields the multiplication rule! zkzl D k C l k zkCl : A multiplicative structure of this type is called a polynomial ring with divided powers. With coefﬁcient ring Q one obtains a polynomial ring H.S nC1I Q/ Š QŒz1. Let n be odd. Then z1z2k D z2kC1, z1z2kC1 D 0, and zk Again use induction and the derivation property. Since z2 1 D 0. Then ‚.z1z2k/ D ‚.z1/z2k z1‚.z2k/ D z2k z1z2k1 D z2k D ‚.z2kC1/, hence z1z2k D z2kC1, since ‚ is an isomorphism. Next compute z1z2kC1 D z1.z1z2k/ D z2 1 z2k D 0. For the last formula use that ‚ ı ‚ is a derivation of even
degree which maps z2k to z2k2. The induction runs then as for even n. The elements z2k generate a polynomial algebra with divided powers and z1 generates an exterior algebra. 2 D kŠz2k. 1 D z2 1, one has z2 17.8 The Theorem of Leray and Hirsch The theorem of Leray and Hirsch determines the additive structure of the cohomology of the total space of a ﬁbration as the tensor product of the cohomology of the base and the ﬁbre. We work with singular cohomology with coefﬁcients in the ring R. A relative ﬁbration.F; F 0/!.E; E0/! B consists of a ﬁbration p W E! B such that the restriction p0 W E0! B to the subspace E0 of E is also a ﬁbration. The ﬁbres of p and p0 over a base point 2B are F and F 0. The case E0 D ; and hence F 0 D ; is allowed. We assume that B is path connected. p (17.8.1) Theorem (Leray–Hirsch). Let.F; F 0/! B be a relative ﬁbration. Assume that H n.F; F 0/ is for each n a ﬁnitely generated free R-module. i!.E; E0/ 428 Chapter 17. Cohomology Let cj 2 H.E; E0/ be a family of elements such that the restrictions i.cj / form an R-basis of H.F; F 0/. Then L W H.B/ ˝ H.F; F 0/! H.E; E0/; b ˝ i.cj / 7! p.b/ Y cj is an isomorphism of R-modules. Thus H.E/ is a free graded H.B/-module with basis fcj g. We explain the statement of the theorem. The source of L is a direct sum of modules H k.B/ ˝ H l.F; F 0/. The elements i c which are contained in H l.F; F 0/ are a ﬁnite R-basis of the R-module H l.
F; F 0/. A basic property of the tensor product says that each element has a unique expression of the form P b ˝ i.c/; b 2 H k.B/: By the conventions about tensor products of graded modules, L is a map of degree zero between graded modules. Proof. Let A B be a subspace. We have the restricted ﬁbrations.F; F 0/!.EjA; E0jA/! A with EjA D p1.A/ and E0jA D E0 \ EjA. The elements cj yield by restriction elements cj jA 2 H.EjA; E0jA/ which again restrict to a basis of H.F; F 0/. We ﬁrst prove the theorem for CW-complexes B by induction over the skeleta B n. If B 0 D fg, then L has the form H 0.B 0/ ˝ H k.F; F 0/! H k.F; F 0/ and it is an isomorphism by the unit element property of the cup product. Suppose the theorem holds for the.n1/-skeleton B n1. We write B n D U [V, where U is obtained from B n by deleting a point in each open n-cell and V is the union of the open n-cells. We use the MV-sequence of U; V and EjU; EjV and obtain a commutative diagram H.U [ V / ˝ M LU [V H.EjU [ V ; E 0jU [ V / H.U / ˝ M ˚ H.V / ˝ M LU ˚LV H.EjU; E0jU / ˚ H.EjV; E0jV / H.U \ V / ˝ M LU \V H.EjU \ V ; E 0jU \ V /: The left column is the tensor product of the MV-sequence for U; V with the graded module M D H.F; F 0/. It is exact, since the tensor product with a free module preserves exactness. We show that LU ; LV and LU \V are isomorphisms. The Five Lemma then shows that LU [V is an isomorphism. This ﬁnishes
the induction step. 17.8. The Theorem of Leray and Hirsch 429 Case U. We have the commutative diagram H.U / ˝ H.F; F 0/ LU H.EjU; E0jU / H.B n1/ ˝ H.F; F 0/ H.EjB n1; E0jB n1/: Since B n1 U, EjB n1 EjU, and E0jB n1 E0jU are deformation retracts, the vertical maps are isomorphisms. We use the induction hypothesis and see that LU is an isomorphism. Case V. The set V is the disjoint union of the open n-cells V D obtain a commutative diagram ` j en j. We H.V / ˝ H.F; F 0/ LV H.EjV; E0jV /.1/ Š Q H.en j / ˝ H.F; F 0/.2/ Š Q H.Ejen j ; E0jen j /.3/ Q H.en j / ˝ H.F; F 0/ Q Len j Q D H.Ejen j ; E0jen j /: Q (1) and (2) are isomorphisms by the additivity of cohomology. The map (3) is the.Mj ˝ N / with a direct sum of homomorphisms of the type ﬁnitely generated free module N and other modules Mj. In a situation like this the tensor product commutes with the product. Hence (3) is an isomorphism. The homomorphisms L.en j / are isomorphisms, since en j is pointed contractible. ˝ N! Q Mj Case U \V. We combine the arguments of the two previous cases. By additivity and ﬁnite generation we reduce to the case of U \en j, a cell with a point deleted. This space has the.n 1/-sphere as a deformation retract. By induction, the theorem holds for an.n 1/-sphere. From the ﬁnite skeleta we now pass to arbitrary CW-complexes via the lim– lim1-sequence (17.1.6). For general base spaces B we pull back the ﬁbration
along a CW-approximation. (17.8.2) Example. Consider the product ﬁbration p D prB W B.F; F 0/! B. Let H.F; F 0/ be a free R-module with homogeneous basis.dj j j 2 J /, ﬁnite in each dimension. Let cj D pr F dj. Then i cj D dj. Therefore (17.8.1) says in this case that H.B/ ˝ H.F; F 0/! H.B.F; F 0//; x ˝ y 7! x y 430 Chapter 17. Cohomology is an isomorphism (of graded algebras). This is a special case of the Künneth Þ formula. (17.8.3) Remark. The methods of proof for (17.8.1) (induction, Mayer–Vietoris sequences) gives also the following result. Suppose H k.F; F 0/ D 0 for k < n. Þ Then H k.E; E0/ D 0 for k < n. Let now h be an arbitrary additive and multiplicative cohomology theory and.F; F 0/!.E; E0/! B a relative ﬁbration over a CW-complex. We prove a Leray- Hirsch theorem in this more general situation. We assume now that there is given a ﬁnite number of elements tj 2 hn.j /.E; E0/ such that the restrictions tj jb 2 hn.j /.Eb; E0 b/ to each ﬁbre over b are a basis of the graded h-module h.Eb; E0 (17.8.4) Theorem (Leray–Hirsch). h.E; E0/ is a free left h.B/-module with basis.tj /. b/. Under these assumptions: Proof. Let us denote by h.C /ht i the free graded h.C /-module with (formal) basis tj in degree n.j /. We have theh.C /-linear map of degree zero '.C / W h.C /ht i!h.EjC; E0jC / which sends tj to tj jC. These maps are natural in
the variable C B. We view h./ht i as a cohomology theory, a direct sum of the theories h./ with shifted degrees. Thus we have Mayer–Vietoris sequences for this theory. If U and V are open in B, we have a commutative diagram of MV-sequences. h.U [ V /ht i '.U [V / h.EjU [ V; E 0jU [ V / h.U /ht i ˚h.V /ht i '.U /˚'.V / h.EjU; E0jU / ˚ h.EjV; E0jV / h.U \ V /ht i '.U \V / h.EjU \ V; E 0jU \ V / We use this diagram as in the proof of (17.8.1). We need for the inductive proof that '.e/ is an isomorphism for an open cell e. This follows from two facts: (1) '.P / is an isomorphism for a point P D fbg B, by our assumption about the tj. (2).EP ; E0 P /!.Eje; E 0je/ induces an isomorphism in cohomology, since EP Eje is a homotopy equivalence by the homotopy theorem for ﬁbrations. The ﬁniteness of the set ftj g is used for the compatibility of products and ﬁnite sums. The passage from the skeleta of B to B uses again (17.1.6). There is a similar application of (17.8.4) as we explained in (17.8.2). 17.9. The Thom Isomorphism 431 17.9 The Thom Isomorphism We again work with a cohomology theory which is additive and multiplicative. Under the obvious ﬁniteness conditions (e.g., ﬁnite CW-complexes) additivity is not needed. Let.p; p0/ W.E; E 0/! B be a relative ﬁbration over a CW-complex. A Thom class for p is an element t D t.p/ 2 hn.E; E0/ such that the restriction to each ﬁbre tb 2 hn.Fb; F 0 b/. We apply the theorem of Ler
ay–Hirsch (17.8.4) and obtain: b/ is a basis of the h-module h.Fb; F 0 (17.9.1) Theorem (Thom Isomorphism). The Thom homomorphism ˆ W hk.B/! hkCn.E; E0/; b 7! p.b/ Y t is an isomorphism. Let us further assume that p induces an isomorphism p W H.B/! H.E/. We use the Thom isomorphism and the isomorphism p in order to rewrite the exact sequence of the pair.E; E0/; we set D ˆ1ı. hk1.E0/ hk.E; E0/ hk.E/ hk.E0/ ı *%%%%%%%%%% j J ˆŠ p hkn.B/ hk.B/ ’’’’’’’’’.p0/ p Let e D e.p/ 2 H n.B/ be the image of t under hn.E; E0/ hn.B/. We call e the Euler class of p with respect to t. From the deﬁnitions we verify that J is the cup product with e, i.e., J.x/ D x Y e. j! hn.E/ (17.9.2) Theorem (Gysin Sequence). Let.E; E0/! B be a relative ﬁbration as above such that p W h.B/ Š h.E/ with Thom class t and associated Euler class e 2 hn.B/. Then we have an exact Gysin sequence! hk1.E0/! hkn.B/.1/! hk.B/.2/! hk.E0/! hknC1.B/! :.1/ is the cup product x 7! x Y e and.2/ is induced by p0. We discuss the existence of Thom classes for singular cohomology H.I R/. In this case it is not necessary to assume that B is a CW-complex (see (17.8.1)). We have for each path w W I! B from b to c a ﬁbre transport w# de�
��ned as follows: Let q W.X; X 0/! I be the pullback of p W.E; E0/! B along w. Then we have isomorphisms induced by the inclusions w# W H n.Fc; F 0 c/ Š H n.X; X 0/ Š! H n.Fb; F 0 b/: * 432 Chapter 17. Cohomology The homomorphism w# only depends on the class of w in the fundamental groupoid. In this manner we obtain a transport functor “ﬁbre cohomology”. (17.9.3) Proposition. We assume that H.Fb; F 0 basis element in H n.Fb; F 0 functor is trivial. bI R/ is a freeR -module with a bI R/. A Thom class exists if and only if the transport c D i Proof. Let t be a Thom class and w W I! B a path from b to c. Then w# ıi b, where ib denotes the inclusion of the ﬁbre over b. Thus w# sends the restricted Thom class to the restricted Thom class and is therefore independent of the path. (In general, the transport is trivial on the image of the i c.) Let now the transport functor be trivial. Then we ﬁx a basis element in a particular ﬁbre H n.Eb; E0 b/ and transport it to any other ﬁbre uniquely (B path connected). A Thom class tC 2 H n.EjC; E0jC / for C B is called distinguished, if the restriction to each ﬁbre is the speciﬁed basis element. This requirement determines tC. By a MV-argument and (17.8.3) we prove by induction that H k.EjB n; E0jB n/ D 0 for k < n and that a distinguished Thom class exists. Then we pass to the limit and to general base spaces as in the proof of (17.8.1). The preceding considerations can be applied to vector bundles. Let W E! B be a real n-dimensional vector bundle and E0 the complement of the zero section. Then for each ﬁbre H n.Eb; E0 b / Š R and is a homotopy equivalence. A Thom class
t./ 2 H n.E; E0I R/ is called an R-orientation of. If it exists, we have a Thom isomorphism and a Gysin sequence. We discuss the existence of Thom classes and its relation to the geometric orientations. (17.9.4) Theorem. There exists a Thom class of with respect to singular cohomology H.I Z/ if and only if the bundle is orientable. The Thom classes with respect to H.I Z/ correspond bijectively to orientations. Proof. Let us consider bundles over CW-complexes. Let t be a Thom class. Consider a bundle chart'W U Rn! 1.U / over a path connected open U. The image of tjU in H n.U.Rn; Rn X0// Š H 0.U / under'and a canonical suspension isomorphism is an element ".U / which restricts to ".u/ D ˙1 for each point u 2 U, and u 7! ".u/ is constant, since U is path connected. We can therefore change the bundle chart by an automorphism of Rn such that ".u/ D 1 for each u. Bundle charts with this property yield an orienting bundle atlas. Conversely, suppose has an orienting atlas. Let'W U Rn! 1.U / be a positive chart. From a canonical Thom class for U Rn we obtain via'a local Thom class tU for 1.U /. Two such local Thom classes restrict to the same Thom class over the intersection of the basic domains, since the atlas is orienting. We can now paste these local classes by the Mayer–Vietoris technique in order to obtain a global Thom class. 17.9. The Thom Isomorphism 433 As a canonical generator of H n.Rn; Rn X 0I Z/ we take the element e.n/ satisfying the Kronecker pairing relation he.n/; en i D1 where en 2 Hn.Rn; Rn X 0I Z/ is the n-fold product e1 e1 of the canonical generator e1 2 H1.R1; R1 X 0I Z/. If a bundle is oriented and 'b W Rn! b a positive isomorphism, then we require b t./ D e.n/ for its associated Thom class t./.'Let t./ 2
hn.E; E0/ be a Thom class of W E! B and e./ 2 hn.B/ the associated Euler class, deﬁned as the restriction of t./ to the zero section t./ 2 hn.E; E0/! hn.E/ s! hn.B/ 3 e./: Here is a geometric property of the Euler class: (17.9.5) Proposition. Suppose has a section which is nowhere zero. Then the Euler class is zero. Proof. Let s0 W B! E0 be a map such that ı s0 D id. The section s0 is homotopic to the zero section by a linear homotopy in each ﬁbre. Therefore e./ is the image of t./ under a map hn.E; E0/! hn.E/! hn.E0/ s0! hn.B/ and therefore zero. The Thom classes and the Euler classes have certain naturality properties. Let f W! be a bundle map. If t./ is a Thom class, then f t./ is a Thom class for and f e./ is the corresponding Euler class. If W X! B and W Y! C are bundles with Thom classes, then the -product t./ t./ is a Thom class for and e./ e./ is the corresponding Euler class. In general, Thom classes are not unique. Let us consider the case of a trivial bundle D pr2 W Rn B! B. It has a canonical Thom class pr 2 en. If t./ is an arbitrary Thom class, then it corresponds under the suspension isomorphism h0.B/! hn.RnB; Rn 0 B/ to an element v./ with the property that its restriction to each point b 2 B is the element ˙1 2 h0.b/. Under reasonable conditions, an element with this property (call it a point-wise unit) is a (global) unit in the ring h0.B/. We call a Thom class for a numerable bundle strict if the restrictions to the sets of a numberable covering correspond under bundle charts and suspension isomorphism to a unit in h0. (17.9.6) Proposition. Let U be a numerable covering of X. Let " 2 h0.X/ be an element such that its restriction to each U 2 U is
a unit. Then " is a unit. Proof. Let X D U [ V and assume that.U; V / is excisive. Let "jU D "U and "jV D "V be a unit. Let U ; V be inverse to "U ; "V. Then U and V have the same restriction to U \ V. By the exactness of the MV-sequence there exists 434 Chapter 17. Cohomology 2 h0.X/ with jU D U ; jV D V. Then x D " 1 has restriction 0 in U and V. Let xU 2 h0.X; U / be a pre-image of x and similarly xV 2 h0.X; V /. Then xU xV D 0 and hence x2 D 0. The relation 1 D ".2 "/ shows that " is a unit. Restrictions of units are units. By additivity, if X is the disjoint union of U and "jU is a unit for each U 2 U, then " is a unit. We ﬁnish the proof as in the proof of (17.9.7). The Thom isomorphism is a generalized (twisted) suspension isomorphism. It is given by the product with a Thom class. Let W E! B be an n-dimensional real vector bundle and t./ 2 hn.E; E0/ a Thom class with respect to a given multiplicative cohomology theory. The Thom homomorphism is the map ˆ./ W hk.B; A/! hkCn.E./; E0./ [ E.A//; where W hk.B; A/! H k.E./; E0.// is the homomorphism induced by. The Thom homomorphism deﬁnes on h.E; E0/ the structure of a left graded h.B/-module. The Thom homomorphism is natural with respect to bundle maps. Let x 7! x t./ D.x/ Y t./ E./ B F f E./ C be a bundle map. Let t./ be a Thom class. We use t./ D F t./ as the Thom class for. Then the diagram hk.C; D/ ˆ./ hkCn.E./; E0./ [ E.D// f F hk.B; A/ ˆ./ h
kCn.E./; E0./ [ E.A// is commutative. We assume that f W.B; A/!.C; D/ is a map of pairs. The Thom homomorphism is also compatible with the boundary operators. Let t.A/ be the restriction of t./. Then the diagram hk.A/ ˆ.A/ hkCn.E.A/; E0.A// Š ı hkC1.B; A/ ˆ./ hkCn.E0./ [ E.A/; E0.A// ı hkCnC1.E./; E0./ [ E.A// is commutative. The Thom homomorphisms are also compatible with the morphisms in the MVsequence. We now consider the Thom homomorphism under a different hypothesis. (17.9.7) Theorem. The Thom homomorphism of a numerable bundle with strict Thom class is an isomorphism. 17.9. The Thom Isomorphism 435 Proof. Let be a numerable bundle of ﬁnite type. By hypothesis, B has a ﬁnite numerable covering U1; : : : ; Ut such that the bundle is trivial over each Uj and the Thom class is strict over Uj. We prove the assertion by induction over t. For t D 1 it holds by the deﬁnition of a strict Thom class. For the induction step consider C D U1 [ [ Ut1 and D D Ut. By induction, the Thom homomorphism is an isomorphism for C, D, and C \ D. Now we use that the Thom homomorphism is compatible with the MV-sequence associated to C, D. By the Five Lemma we see that ˆ./ is an isomorphism over C [ D. Suppose the bundle is numerable over a numerable covering U. Assume that for each U 2 U the Thom class t.U / is strict. In that case ˆ.U / is an isomorphism. For each V U the Thom class t.V / is also strict. There exists a numerable covering.Un j n 2 N/ such that jUn is numerable of ﬁnite type with strict Thom class. Let.n j n 2 N/ be a numeration of.un/. Set f W
B! Œ0; 1Œ, f.x/ D P j >n j.x/ and S n j D1 supp.j /, then 1 D n nn.x/. If x … j 1 j.x/ D P P therefore f.x/ D P j >n jj.x/.n C 1/ S P j >n j.x/ D n C 1: n Hence f 1Œ0; n is contained in j D1 supp.j /. Hence f 1r; sŒ is always contained in a ﬁnite number of Vj and therefore the bundle over such a set is numerable of ﬁnite type. The sets Cn D f 12n 1; 2n C 1Œ are open and disjoint. Over Cn Cn the isomorphism the bundle is of ﬁnite type. By additivity, we have for C D hk.EjC; E0jC / Š hk.EjCn; E0jCn/. The Thom classes over Cn yield a unique Thom class over C. Now we use the same argument for Dn D f 12n; 2n C 2Œ, f 1n; n C 1Œ and then apply the MV-argument D D to C, D. Dn and C \ D D S S Q S (17.9.8) Example. Let 1.n/ W H.1/! CP n be the canonical line bundle introduced in (14.2.6). A complex vector bundle has a canonical orientation and an associated Thom class (17.9.4). Let c 2 H 2.CP n/ be the Euler class of 1.n/. The associated sphere bundle is the Hopf ﬁbration S 2nC1! CP n. Since H k.S 2nC1/ D 0 for 0 < k < 2n C 1 the multiplications by the Euler class c 2 H 2.CP n/ are isomorphisms Z Š H 0.CP n/ Š H 2.CP n/ Š Š H 2n.CP n/ and similarly H k.CP n/ D 0 for odd k. We obtain the structure of the cohomology ring H.CP n/ Š ZŒc=.cnC1/: In the in
ﬁnite case we obtain H.CP 1I R/ Š RŒc where R is an arbitrary commutative ring. 436 Chapter 17. Cohomology A similar argument for the real projective space yields for the cohomology ring H.RP nI Z=2/ Š Z=2Œw=.wnC1/ with w 2 H 1.RP nI Z=2/ and for the inﬁnite Þ projective space H.RP 1I Z=2/ Š Z=2Œw. (17.9.9) Example. The structure of the cohomology ring of RP n can be used to give another proof of the Borsuk–Ulam theorem: There does not exist an odd map F W S n! S n1. For the proof let n 3. (We can assume this after suspension.) Suppose It induces a map f W RP n! RP n1 of the orbit there exists an odd map F. spaces. Let v W I! S n be a path from x to x. Composed with the orbit map pn W S n! RP n we obtain a loop pnv that generates 1.RP n/ Š Z=2. The path u D F v from F.x/ to F.x/ D F.x/ yields a loop that generates 1.RP n1/. Hence f W 1.RP n/! 1.RP n1/ is an isomorphism. This fact implies (universal coefﬁcients) that f is an isomorphism in H 1.I Z=2/. Since wn D 0 in H.RP n1I Z=2/ but wn 6D 0 in H.RP nI Z=2/, we have arrived at a contradicÞ tion. Problems 1. A point-wise unit is a unit under one of the following conditions: (1) For singular cohomology H.I R/. (2) B has a numerable null homotopic covering. (3) B is a CW-complex. 2. Prove the Thom isomorphism for vector bundles over general spaces and for singular cohomology. 3. Let W E./! B and W E./! B be vector bundles with Thom classes t./ and t./. Deﬁne a relative Thom homomorphism as the composition of x 7! x t./
, hk.E./; E0./[E.A//! hkCn.E./E./;.E0./[E.A//E./[E./E0.// with the map induced by.E. ˚ /; E0. ˚ / [ E.A ˚ A//!.E./ E./;.E0./ [ E.A/ E./ [ E./ E 0.//; a kind of diagonal, on each ﬁbre given by.b; v; w/ 7!..b; v/;.b; w//. This is the previously deﬁned map in the case that dim D 0. The product t./ t./ is a Thom class and also its restriction t. ˚ / to the diagonal. Using this Thom class one has the transitivity of the Thom homomorphism ˆ./ˆ./ D ˆ. ˚ /. 4. Given such that ri D id (a retract). Let W E! X be a bundle over X and r D the induced bundle. Let t./ be a Thom class and t./ its pullback. If ˆ./ is an isomorphism, then ˆ./ is an isomorphism. 5. Let Cm S 1 be the cyclic subgroup of m-th roots of unity. A model for the canonical map pm W BCm! BS1 is the sphere bundle of the m-fold tensor product m D ˝ ˝ of the canonical (universal) complex line bundle over BS1. 6. Let R be a commutative ring. Then H.BS1I R/ Š RŒc where c is the Euler class of. 17.9. The Thom Isomorphism 437 7. We use coefﬁcients in the ring R. The Gysin sequence of pm W BCm! BS1 splits into short exact sequences 0! H 2k1.BCm/! H 2k2.BS1/ Ymc! H 2k.BS1/ p m! H 2k.BCm/! 0: This implies H 2k.BCm/ Š R=mR, H 2k1.BCm/ Š mR for k > 0, where mR is the m-torsion h x 2 R j mx D
0 i of R. In even dimensions we have the multiplicative isomorphism H 2.BS1/=.mc/ Š H 2.BCm/ induced by pm. C! BCm has a contractible 8. The sphere bundle of the canonical bundle ECm Cm total space. Therefore the Gysin sequence of this bundle shows that the cup product Yt W H j.BCmI R/! H j C2.BCmI R/ is an isomorphism for j > 0; here t D p 9. The cup product mc. H 1.BCmI R/ H 1.BCmI R/! H 2.BCmI R/ is the R-bilinear form mR mR! R=mR;.u; v/ 7! m.m 1/=2 uv: Here one has to take the product of u; v 2 mR R and reduce it modulo m. Thus if m is odd, this product is zero; and if m is even it is.u; v/ 7! m=2 uv. Chapter 18 Duality We have already given an introduction to duality theory from the view point of homotopy theory. In this chapter we present the more classical duality theory based on product structures in homology and cohomology. Since we did not introduce products for spectral homology and cohomology we will not directly relate the two approaches of duality in this book. Duality theory has several aspects. There is, ﬁrstly, the classical Poincaré duality theorem. It states that for a closed orientable n-dimensional manifold the groups H k.M / and Hnk.M / are isomorphic. A consequence is that the cup product pairing H.M / ˝ H n.M /! H n.M / is a regular bilinear form (say with ﬁeld coefﬁcients). This quadratic structure of a manifold is a basic ingredient in the classiﬁcation theory (surgery theory). The cup product pairing for a manifold has in the context of homology an interpretation as intersection. Therefore the bilinear cup product form is called the intersection form. In the case of a triangulated manifold there exists the so-called dual cell decomposition, and the simplicial chain complex is isomorphic to the cellular cochain complex of dual cells;
this is a very strong form of a combinatorial duality theorem [167]. The second aspect relates the cohomology of a closed subset K Rn with the homology of the complement Rn X K (Alexander duality). This type of duality is in fact a phenomenon of stable homotopy theory as we have explained earlier. Both types of duality are related. In this chapter we prove in the axiomatic context of generalized cohomology theories a theorem which compares the cohomology of pairs.K; L/ of compact subsets of an oriented manifold M with the homology of the dual pair.M X L; M X K/. The duality isomorphism is constructed with the cap product by the fundamental class. We construct the cap product for singular theory. 18.1 The Cap Product The cap product relates singular homology and cohomology with coefﬁcients in the ring R. Let M and N be left R-modules. The cap product consists of a family of R-linear maps H k.X; AI M / ˝ Hn.X; A [ BI N /! Hnk.X; BI M ˝R N /; x ˝ y 7! x Z y and is deﬁned for excisive pairs.A; B/ in X. (Compare the deﬁnition of the cup If a linear map W M ˝ N! P is given, product for singular cohomology.) 18.1. The Cap Product 439 we compose with the induced map; then x Z y 2 Hnk.X; BI P /. This device is typically applied in the cases M D R and W R˝N! N is an R-module structure, or M D N D ƒ is an R-algebra and W ƒ ˝ ƒ! ƒ is the multiplication. We ﬁrst deﬁne a cap product for chains and cochains S k.XI M / ˝ Sn.XI N /! Snk.XI M ˝ N /;'˝ c 7!'Z c: Given'2 S p.XI M / and W pCq D Œe0; : : : ; epCq! X, we set'Z. ˝ b/ D.1/pq.'.
jŒeq; : : : ; epCq ˝ b/jŒe0; : : : ; eq and extend linearly. (Compare in this context the deﬁnition of the cup product.) From this deﬁnition one veriﬁes the following properties. (1) Let f W X! Y be continuous. Then f#.f #' Z c/ D'Z f#c: (2) @.' Z c/ D ı' Z c C.1/j'j Z @c. (3).' /. (4) 1 Z c D c. Case (3) needs conventions about the coefﬁcients. It can be applied in the case that'2 S p.XI R/; 2 S q.XI ƒ/ and c 2 Sn.XI ƒ/ for an R-algebra ƒ. In case (4) we assume that 1 2 S 0.XI R/ is the cocycle which send a 0-simplex to 1 and c 2 Sn.XI N / for an R-module N. We now extend the deﬁnition to relative groups. If'2 S p.X; AI M / S p.XI M / and c 2 SpCq.AI N / C SpCq.BI N /, then'Z c 2 Sq.BI M ˝ N /. Thus we have an induced cap product S p.X; AI M / ˝ SpCq.XI N / SpCq.AI N / C SpCq.BI N /! Sq.XI M ˝ N / Sq.BI M ˝ N / : Let A; B be excisive. We use the chain equivalence S.A/ C S.B/! S.A [ B/. After passing to cohomology we obtain the cap product as stated in the beginning. We list the 18.1.1 Properties of the cap product. (1) For f W.XI A; B/!.X 0I A0; B 0/, x0 2 H p.X 0; A0I M /, and for u 2 HpCq.X; A [ BI N / the relation f.f x0 Z u/ D x0 Z fu
holds. (2) Let A; B be excisive, jB W.B; A \ B/!.X; A [ B/ the inclusion and @B W HpCq.X; A [ B/ @! HpCq1.A [ B; A/ Š HpCq1.B; A \ B/: Then for x 2 H p.X; AI M /, y 2 HpCq.X; A [ BI N /, B x Z @B y D.1/p@.x \ y/ 2 Hq1.BI M ˝ N /: j 440 Chapter 18. Duality (3) Let A; B be excisive, jA W.A; A \ B/!.X; B/ the inclusion and @A W HpCq.X; A [ B/ @! HpCq1.A [ B; B/ Š HpCq1.A; A \ B/: Then for x 2 H p.AI M /, y 2 HpCq.X; A [ BI N /, jA.x Z @Ay/ D.1/pC1ıx Z y 2 Hq1.X; BI M ˝ N /: (4) 1 Z x D x, 1 2 H 0.X/, x 2 Hn.X; B/. (5).x Y y/ Z z D x Z.y Z z/ 2 Hnpq.X; C I ƒ/ for x 2 H.X; AI R/, y 2 H.X; BI ƒ/; z 2 H.X; A [ B [ C I ƒ/. (6) Let " W H0.XI M ˝ N /! M ˝ N denote the augmentation. For x 2 H p.X; AI M /, y 2 Hp.X; AI N /, ".x Z y/ D hx; y i where h; i is the Kronecker pairing. We display again the properties in a table and refer to the detailed description above. f.f x0 Z u/ D x0 Z fu B x Z @B y D.1/jxj@.x Z y/ j.jA/.x Z @Ay/ D.1/jxjC1ıx
Z y 1 Z x D x.x Y y/ Z z D x Z.y Z z/ ".x Z y/ D hx; y i We use the algebra of the cap product and deduce the homological Thom iso- morphism from the cohomological one. (18.1.2) Theorem. Let W E! B be an oriented n-dimensional real vector bundle with Thom class t 2 H n.E; E0I Z/. Then tZ W HnCk.E; E0I N /! Hk.EI N / is an isomorphism. Proof. Let z 2 S n.E; E0/ be a cocycle which represents t. Then the family SnCk.E; E0I N /! Sk.EI N /; x 7! z Z x 18.2. Duality Pairings 441 is a chain map of degree n. This chain map is obtained from the corresponding one for N D Z by taking the tensor product with N. It sufﬁces to show that the integral chain map induces an isomorphism of homology groups, and for this purpose it sufﬁces to show that for coefﬁcients in a ﬁeld N D F an isomorphism is induced (see (11.9.7)). The diagram H k.EI F / Yt H kCn.E; E0I F / ˛Š Hom.Hk.EI F /; F /.tZ/ ˛Š Hom.HkCn.E; E0I F /; F / is commutative (by property (6) in 18.1.1) where ˛ is the isomorphism of the universal coefﬁcient theorem. Since Yt is an isomorphism, we conclude that tZ is an isomorphism. Problems 1. The cap product for an excisive pair.A; B/ in X is induced by the following chain map (coefﬁcient group Z): S.X; A/ ˝ S.X; A [ B/ 1˝ S.X; A/ ˝ S.X/=.S.A/ C S.B// 1˝D! S.X; A/ ˝ S.X; B/
˝ S.X; A/ 1˝! S.X; A/ ˝ S.X; A/ ˝ S.X; B/ "! Z ˝ S.X; B/ Š S.X; B/: D is an approximation of the diagonal, the graded interchange map, and " the evaluation. The explicit form above is obtained from the Alexander–Whitney map D. 2..x y/ Z.a b/ D.1/jyjjaj.x Z a/.y Z b/. 3. From the cap product one obtains the slant product x ˝ u 7! xnu which makes the following diagram commutative: H q.X; A/ ˝ Hn..X; A/.Y; B// n pr ˝1 Hnq.Y; B/ pr H q..X; A/ Y / ˝ Hn..X; A/.Y; B// Z Hnq.X.Y; B//: The properties (1)–(5) of the cap product can be translated into properties of the slant product, and the cap product can be deduced from the slant product. (This is analogous to the Y- and -product.) 18.2 Duality Pairings We use the properties of the cap product in an axiomatic context. Let h be a cohomology theory and k; h homology theories with values in R- MOD. A 442 Chapter 18. Duality duality pairing (a cap product) between these theories consists of a family of linear maps hp.X; A/ ˝ kpCq.X; A [ B/! hq.X; B/; x ˝ y 7! x Z y deﬁned for pairs.A; B/ which are excisive for the theories involved. They have the following properties: (1) Naturality. For f W.XI A; B/!.X 0I A0; B 0/ the relation f.f x0 Z u/ D x0 Z fu holds. (2) Stability. Let A; B be excisive. Deﬁne the mappings jB and @B as in (18.1.1). Then j (3) Stability. Let A; B be excisive. Deﬁne the m
appings jA and @A as in (18.1.1). Then.jA/.x Z @Ay/ D.1/jxjC1ıx Z y. (4) Unit element. There is given a unit element 1 2 k0.P /. The homomorphism hk.P /! hk.P /, x 7! x Z 1 is assumed to be an isomorphism (P a point). B x Z @B y D.1/jxj@.x Z y/. (In the following investigations we deal for simplicity of notation only with the case h D k.) Note that we do not assume given a multiplicative structure for the cohomology and homology theories. As a ﬁrst consequence of the axioms we state the compatibility of the cap product with the suspension isomorphisms. (18.2.1) Proposition. The following diagrams are commutative: hp.X; A/ ˝ hpCq.X; A [ B/ Z hq.X; B/ pr ˝ hp.IX; IA/ ˝ hpCqC1.IX; @IX [ IA [ IB/ Z.1/p hqC1.IX; @IX [ IB/, hp.X; A/ ˝ hpCq.X; A [ B/ Z hq.X; B/ ˝.1/p pr hpC1.IX; @IX [ IA/ ˝ hpCqC1.IX; @IX [ IA [ IB/ Z hq.IX; IB/. (For the second diagram one should recall our conventions about the suspension isomorphisms, they were different for homology and cohomology. Again we use notations like IX D I X.) Proof. We consider the ﬁrst diagram in the case that A D ;. The proof is based on 18.2. Duality Pairings 443 the next diagram. hp.1X / ˝ hpCq.1X; 1B/ Z hq.1X; 1B/ j ˝j 1 j 1 hp.@IX [ IB/ ˝ hpCq.@IX [ IB; IB/ Z hq.@IX [ IB; IB/ 1˝i i hp.@IX [ IB/ ˝ hpCq.@IX [ IB; IB
/ Z hq.@IX [ IB/ k˝@ @ hp.IX/ ˝ hpCqC1.IX; @IX [ IB/ Z hqC1.IX; @IX [ IB/ The maps i, j, k are inclusions. The ﬁrst and the second square commute by naturality. The third square is.1/p-commutative by stability (2). (18.2.2) Proposition. Let en 2 hn.Rn; Rn X 0/ be obtained from 1 2 h0.P / under an iterated suspension isomorphism. Then hk.Rn/! hnk.Rn; Rn X 0/; x 7! x Z en is for k 2 Z and n 1 an isomorphism. Proof. This follows by induction on n. One uses the ﬁrst diagram in (18.2.1) and an analogous suspension isomorphism with.R; R X 0/ in place of.I; @I /. (18.2.3) Proposition. Let.U; V / and.U 0; V 0/ be pairs of open subsets in the space X D U [ U 0. Let 2 hn.U [ U 0; V [ V 0/ be a ﬁxed element. From it we produce elements ˛ and ˇ via 2 hn.X; V [ V 0/! hn.X; V [ U 0/ Š hn.U; V [ U U 0/ 3 ˛; 2 hn.X; V [ V 0/! hn.X; U [ V 0/ Š hn.U 0; V 0 [ U U 0/ 3 ˇ: Then the diagram hk1.U; V / Z ˛ hk1.U U 0; V U 0/ ı hk.U 0; U U 0/ Z ˇ hnkC1.U; U U 0/ @ hnkC1.U U 0; U V 0/ hnk.U 0; V 0/ is commutative. (We again have used notations like U U 0 D U \ U 0.) 444 Chapter 18. Duality Proof. We use naturality and stability (2) and show that the down-right path
of the diagram is.1/k1 times the map hk1.U; V /! hk1.U U 0; V U 0/ Z ˛1 hnk.U U 0; U V 0/! hnk.U 0; V 0/ and the right-down path.1/k times the analogous map where ˛1 is replaced by ˇ1; the element ˛1 is obtained from ˛ via the morphism ˛ 2 hn.U; V [ U U 0/ @! hn1.V [ U U 0; V / Š hn1.U U 0; V U 0/! hn1.U U 0;.U V 0/ [.V U 0// 3 ˛1 and ˇ1 from ˇ via the analogous composition in which the primed and unprimed spaces are interchanged. Thus it remains to show ˛1 D ˇ1. This is essentially a consequence of the Hexagon Lemma. One of the outer paths in the hexagon is given by the composition hn.U [ U 0; V [ V 0/! hn.U [ U 0; U [ V 0/ Š hn.U 0 [ V;.V [ U 0/.U [ V 0// @! hn1..U [ V 0/.V [ U 0/; V [ V 0/ and the other path is obtained by interchanging the primed and unprimed objects. The center of the hexagon is hn.U [ U 0;.U [ V 0/.V [ U 0//. We then compose the outer paths of the hexagon with the excision hn1.U U 0; U V 0 [ V U 0/! hn1..U [ V 0/.V [ U 0/; V [ V 0/I then is mapped along the paths to ˛1 and ˇ1, respectively; this follows from the original deﬁnition of the elements by a little rewriting. The displayed morphism yields ˇ1. 18.3 The Duality Theorem For the statement of the duality theorem we need two ingredients: A homological orientation of a manifold and a duality homomorphism. We begin with the former. Let M be an n-dimensional manifold. For K L M we write r
L K W h.M; M X L/! h.M; M X K/ for the homomorphism induced by the inclusion, and r L x in the case that K D fxg. An element oL 2 hn.M; M X L/ is said to be a homological orientation along L if for each y 2 L and each chart'W U! Rn centered at y the image of oL under hn.M; M X L/ r L y hn.M; M X y/ Š hn.U; U X y/'hn.Rn; Rn X 0/ 18.3. The Duality Theorem 445 is ˙en where en is the element which arises from 1 2 h0 under suspension. A family.oK j K M compact/ is called coherent if for each compact pair K L the restriction relation r L K oL D oK holds. A coherent family.oK/ of orientations is a (homological ) orientation of M. If M is compact, then K D M is allowed and an orientation is determined by the element oM 2 hn.M /, called the fundamental class of M. In order to state the duality theorem we need the deﬁnition of a duality homomorphism. We ﬁx a homological orientation.oK/ of M. Given closed sets L K M and open sets V U M such that L V; K U. We ﬁx an element z 2 hn.M; M X K/. From oK D z we obtain zU V z 2 hn.M; M X K/ KL via hn.M;.M X K/ [ V / Š.#/ zU V KL 2 hn.U X L;.U X K/ [.V X L//: The morphism.#/ is an excision, since M X.U X L/ D.M X U / [ L (closed) is contained in.M X K/ [ V (open). From zU V KL we obtain the homomorphism DU V KL via the commutative diagram hk.U; V / DU V KL hk.U X L; V X L/ Z zU V KL hnk.M X L; M X K/ Š hnk.U X L; U X K
/: We state some naturality properties of these data. They are easy consequences of the naturality of the cap product. (18.3.1) Lemma. Let.K; L/.U 0; V 0/.U; V / and i W.U 0 X L; U 0 X K; V 0 X L/.U X L; U X K; V X L/: Then izU 0V 0 KL D zU V KL and DU V KL D DU 0V 0 KL ı i. (18.3.2) Lemma. Let.K0; L0/.K; L/.U; V / and j W.U X L; U X K; V X L/.U X L0; U X K0; V X L0/: Then jzU V KL D zU V K0L0 and j ı DU V K0L0 D DU V KL. The naturality (18.3.1) allows us to pass to the colimit over the neighbourhoods.U; V / of.K; L/ in M. We obtain a duality homomorphism DKL W Lhk.K; L/ D colimU V hk.U; V /! hnk.M X L; M X K/: 446 Chapter 18. Duality We explain this with some remarks about colimits. An element x 2 hk.U; V / represents an element of the colimit. Two elements x 2 hk.U; V / and x0 2 hk.U 0; V 0/ represent the same element if and only if there exists a neighbourhood.U 00; V 00/ with U 00 U \ U 0, V 00 V \ V 0 such that x and x0 have the same restriction in hk.U 00; V 00/. Thus we have canonical homomorphisms lU V W hk.U; V /! Lhk.K; L/. Via these homomorphisms the colimit is characterized by a universal property: If U V W hk.U; V /! h is a family of homomorphisms such that U 0V 0 ı i D U V for the restrictions i W hk.U; V /! hk.U 0; V 0/, then there exists a unique homomorphism W Lhk
.K; L/! h such that lU V D U V. The restrictions hk.U; V /! hk.K; L/ are compatible in this sense, and we obtain a canonical homomorphism Lhk.K; L/! hk.K; L/. In sufﬁciently regular situations this homomorphism is an isomorphism; we explain this later. (18.3.3) Duality Theorem. Let M be an oriented manifold. Then the duality homomorphism DKL is, for each compact pair.K; L/ in M, an isomorphism. We postpone the proof and discuss some of its applications. Let M be compact and ŒM 2 hn.M / a fundamental class. In the case.K; L/ D.M; ;/ we have Lhk.M; ;/ D hk.M / and DKL is the cap product with ŒM. Thus we obtain as a special case: (18.3.4) Poincaré Duality Theorem. Suppose the compact n-manifold is oriented by the fundamental class ŒM 2 hn.M /. Then hk.M /! hnk.M /; x 7! x Z ŒM is an isomorphism. A duality pairing exists for the singular theory H p.X; AI G/ ˝ HpCq.X; A [ BI R/! Hq.X; BI G/ for commutative rings R and R-modules G. The Euclidean space Rn is orientable for H.I Z/. Thus we have: (18.3.5) Alexander Duality Theorem. For a compact pair.K; L/ in Rn LH k.K; LI G/ Š Hnk.Rn X K; Rn X LI G/: A similar isomorphism exists for S n in place of Rn. (18.3.6) Example. We generalize the Jordan separation theorem. Let M be a connected and orientable (with respect to H.I Z/) n-manifold. Suppose that H1.M I R/ D 0. Let A M be compact, A 6D M. Then LH n1.AI R/ is a free R-module, and j0.M X A/j D1 C
rank LH n1.AI R/. 18.4. Euclidean Neighbourhood Retracts 447 By duality LH n1.AI R/ Š H1.M; M X AI R/. The hypothesis shows @ W H1.M; M X AI R/ Š QH0.M X AI R/; and the latter is a free R-module of rank j0.M X A/j 1. Þ (18.3.7) Example. H 2.RP 2I Z/ Š Z=2. This is not a free Z-module. Hence the projective plane cannot be embedded into S 3. (A similar proof shows that RP 2n Þ has no embedding into S 2nC1.) (18.3.8) Remark. From Alexander duality and the Thom isomorphism one can deduce Poincaré duality. Let M RnCt be a smooth closed submanifold of dimension n. Suppose we have an Alexander duality isomorphism hk.M / Š hnCt k.RnCt ; RnCt X M /. Let W E./! U be a tubular map. We use and excision and obtain hnCtk.RnCt ; RnCt X M / Š hnCtk.E./; E0.//. Suppose the normal bundle is oriented by a Thom class. Then we have a Thomisomorphism hnCtk.E./; E0.// Š hnk.M /. Altogether we obtain an isomorphism hk.M / Š hnk.M / of Poincaré duality type. A similar device works if we start from an isomorphism hk.M / Š hnCkt.RnCt ; RnCt X M / and use a homological Thom isomorphism. This approach would be used if one starts with Þ homotopical duality as a foundation stone. Problems 1. Let D R2 be connected and open. The following are equivalent: (1) D is homeomorphic to R2. (2) D is simply connected. (3) H1.DI Z/ D 0. (4) H 1.DI Z/ D 0. (5) R2 X D is connected. (6) The boundary of D is connected. (7) If
J D is a Jordan curve, then D contains the interior of J. [44, p. 394 ] 2. Let i W S n! K.Z; n/ be an inclusion of a subcomplex which induces an isomorphism of n. For each compact subset K RnC1 the induced map i W ŒK; S n! ŒK; K.Z; n/ is bijective. 3. Use cohomology H n.XI Z/ D ŒX; K.Z; n/ deﬁned with an Eilenberg–Mac Lane complex K.Z; n/. Then for a compact subset X in a Euclidean space LH n.XI Z/ Š H n.XI Z/. Similar isomorphisms hold for the stable cohomotopy groups. 4. Rn is orientable for each homology theory. Let K D.r/ D fx j kxk r g be compact. Deﬁne oK as the image of the canonical class under hn.Rn; RnX0/ hn.Rn; RnXD.r//! hn.Rn; Rn X K/. 18.4 Euclidean Neighbourhood Retracts For applications it is interesting to compare LH k with H k. 448 Chapter 18. Duality A space X is called a Euclidean neighbourhood retract (ENR ) if there exists an embedding j W X! Rn, an open neighbourhood U of j.X/ in Rn and a retraction r W U! X, i.e., jr D id.X/. Let X Rn be a retract of an open set U, then X is closed in U and hence locally compact: Let r W U! X be a retraction; then X is the coincidence set of r W U! U and id W U! U. Recall that a locally compact set Y in a Hausdorff space Z is locally closed, i.e., has the form Y D xY \ W for an open set W in Z. (18.4.1) Proposition. Let X Rn be a retract of an open neighbourhood. Let Z be a metric space and Y Z homeomorphic to X. Then Y is a retract of an open neighbourhood V of Y in Z. Proof. Let f W X! Y be a homeomorphism and r W U!
X a retraction. Then Y is locally compact and we can write Y D xY \ W with an open W Z. Then Y is f 1! X! Rn has a continuous closed in W. Since W is a normal space, the map Y extension h W W! Rn by the Tietze extension theorem. Let V D h1.U /. Then f rh W V! Y is a retraction of Y V. (18.4.2) Proposition. Let X Rn be locally compact. Then there exists an embedding of X into RnC1 as a closed subset. Proof. Let U Rn be open (U 6D Rn). Then j W U! Rn R; x 7!.x; d.x; Rn X U /1/ is an embedding. The image of j is closed, since j.U / D f.x; t/ j t d.x; Rn X U / D 1g: We can assume that X U is closed; then j.X/ is closed in j.U /, hence closed in RnC1. (18.4.3) Proposition. Let X be an ENR. Suppose f0; f1 W Y! X are maps which coincide on a subset B Y. Then there exists a neighbourhood W of B in Y and a homotopy h W f0jW'f1jW relative to B. Proof. Let X i! U r! X be a presentation as a retract with U Rn open. Let W D fy j.1 t/if0.y/ C tif1.y/ 2 U for all t 2 I g: Then certainly B W. Since W Y I! Rn;.y; t/ 7!.1 t/if0.y/ C tif1.y/ is continuous, 1.U / is open. If fyg I 1.U /, then there exists an open neighbourhood Uy of y such that Uy I 1.U /. Hence W is open in Y. A suitable homotopy h is now obtained as the restriction of to W I. 18.4. Euclidean Neighbourhood Retracts 449 (18.4.4) Remark. Suppose B X are Euclidean neighbourhood retracts. Then there exists a retraction r W V! B from an open neighbourhood V of B in X
. There further exists a neighbourhood W V of B in X such that j W W V and i ı.rjW / W W! B V are homotopic relative to B. An ENR is locally contractible: Each neighbourhood V of x contains a neighbourhood W of x such that W V is homotopic to W! fxg V relative Þ to fxg. (18.4.5) Lemma. Let the Hausdorff space X D X1 [ [ Xr be a union of locally compact open subsets Xj which are homeomorphic to a subset of a Euclidean space. Then X is homeomorphic to a closed subset of a Euclidean space. Proof. There exist embeddings hi W Xi! Rm.i/ as a closed subset. We extend hi to a continuous map ki W X! S m.i/ D Rm.i/ [ f1g by ki.X X Xi / D f1g (if Q r iD1 S m.i/ is an embedding. The product of X 6D Xi ). The product.ki / W X! Q Rm.i/C1, and then we can apply (18.4.2) if the spheres can be embedded into i necessary. (18.4.6) Theorem. Let the Hausdorff space X D X1 [ [ Xr be a union of open subsets Xi which are ENR’s. Then X is an ENR. Proof. Induction on r. It sufﬁces to consider X D X0 [ X1. By (18.4.5) we can assume that X is a closed subset of some Rn. Let ri W Ui! Xi be retractions (see (18.4.1)). Set U01 D r 1 0.X0 \ X1/ \ r 1 1.X0 \ X1/: Then r0; r1 W U01! X0 \ X1 are retractions of a neighbourhood. The open subset X0 \ X1 of the ENR X0 is an ENR. Hence there exists X0 \ X1 V01 U01 such that r0; r1 are homotopic on V01 relative to X0 \ X1 by a homotopy rt (see (18.4.3)). Let V0 U0; V1 U1 be
open neighbourhoods of X X X1; X X X0 such that xV0 \ xV1 D ;. Choose a continuous function W Rn! Œ0; 1 such that.V0/ D 0 and.V1/ D 1. Let V D V0 [ V01 [ V1. Then W V! X, deﬁned as jV0 D r0jV0, jV1 D r1jV1,.x/ D r.x/.x/ for x 2 V01 is a suitable retraction. (18.4.7) Corollary. A compact manifold is an ENR. (18.4.8) Remark. Since an ENR is dominated by a CW-complex it has the homotopy type of a CW-complex. A compact ENR is dominated by a ﬁnite CW-complex; therefore its singular homology groups with coefﬁcients in Z are ﬁnitely generated Þ abelian groups. This holds in particular for compact manifolds. (18.4.9) Proposition. Let K be a compact ENR in an n-manifold M. Then the canonical map W LH k.K/! H k.K/ is an isomorphism. 450 Chapter 18. Duality Proof. We use that M is a metrizable space or, at least, that open subsets are normal. Then K is a retract r W U! K of an open neighbourhood U of K in M. Suppose x 2 H k.K/; then r.x/ represents an element Lx 2 LH k.K/ with. Lx/ D x. This shows that is surjective. Let xU 2 H k.U / represent an element in the kernel of. Suppose there exists a neighbourhood V U of K and a homotopy from j W V U to ir W V! K! U. Then we have the situation xU H k.U / j xV H k.V / i # ###### j H k.K/ r H k.V /: Since i.xU / D 0, by assumption, we see that xV D 0; but xV represents the same element as xU. The homotopy exists by (18.4.3) if also U is an ENR. If we choose U as the union of a ﬁn
ite number of sets which are homeomorphic to open subsets of Rn, then we can apply (18.4.6). Let X be an ENR and X Rn. Since the canonical homomorphism Lh.X/! h.X/ is an isomorphism, Lh.X/ does not depend on the embedding X Rn. Let M be a compact n-manifold. Then Hk.M I Z/ and H k.M I Z/ are ﬁnitely generated abelian groups. For each ﬁeld F the groups Hk.M I F / are ﬁnite-dimensional vector spaces. The Euler characteristic.M I F / is independent of F and equal to the Euler characteristic.M /. Let K M be a compact ENR. Then H ni.KI F2/ Š LH ni.KI F2/ Š Hi.M; M X KI F2/ and these are ﬁnite-dimensional vector spaces over the prime ﬁeld F2. (18.4.10) Proposition. H.KI F2/ is ﬁnite-dimensional if and only if the same holds for H.M X KI F2/. If ﬁniteness holds, then for the Euler characteristic 2 the relation 2.M / D 2.M X K/ C.1/n2.K/ holds. (Note that 2.K/ D.K/.) Proof. The ﬁrst statement follows from the exact homology sequence of the pair.M; M X K/. It also yields 2.M / D 2.M X K/ C 2.M; M X K/. The equality 2.M; M X K/ D.1/n2.K/ is obtained, if we insert the consequence dim Hi.M; M X KI F2/ D dim H ni.KI F2/ of the duality into the homological deﬁnition of the Euler characteristic. # * * 18.5. Proof of the Duality Theorem 451 (18.4.11) Corollary. Let M be a closed manifold of odd dimension. Then.M / D 0. IfK M is a compact ENR, then.K/ D.M X K/. Problems 1. Let F be a compact, connected, non
-orientable surface. The universal coefﬁcient formula and H1.F I Z/ Š Zg1 ˚ Z=2 show H 2.F I Z/ Š Z=2. Therefore F cannot be embedded into S 3. 2. Let S R2 be the pseudo-circle. Show, heuristically, that the pseudo-circle has a system of neighbourhoods U1 U2 with Ui Š S 1 Œ0; 1 and \Un D S. Then LH 1.SI Z/ Š H1.R2; R2 X SI Z/ Š QH0.R X SI Z/ Š Z; the last isomorphism because R2 X S has two path components. By the universal coefﬁcient formula H 1.SI Z/ Š Hom.H1.S/; Z/. The singular homology group H1.SI Z/ is zero, a singular 1-chain is always contained in a contractible subset. In fact, S has the weak homotopy type of a point. This shows that singular theory is the wrong one for spaces like S. 3. Let F R3 be a connected orientable compact surface. Then R3 X F has two path components (interior and exterior). 4. Let M D RnC1 and S RnC1 be homeomorphic to S n. Then S is an ENR and LH n.SI Z/ Š H n.SI Z/ Š Z. From the duality theorem one obtains that RnC1 X S has two path components. 18.5 Proof of the Duality Theorem We have to collect some formal properties of the groups Lh and of the duality homomorphisms D. We want the Lhk.K; L/ to be part of functors from the category K.M / of compact pairs in M and inclusions. Let.K0; L0/.K; L/. The induced map Lhk.K; L/! Lhk.K0; L0/ sends an element represented by x 2 hk.U; V / to the element which is represented by the same x. This makes the Lhk into functors on K.M / and the canonical maps Lhk! hk into natural transformations. We deﬁne a coboundary operator Lı W Lhk.L
/! LhkC1.K; L/ as follows. Let V L be open. Choose U V as an open neighbourhood of K. Then we map representing elements via ı W hk.V /! hkC1.U; V /. This process yields a well-deﬁned Lı and the diagram Lhk.L/ hk.K/ Lı ı LhkC1.K; L/ hkC1.K; L/ is commutative. From (18.3.2) we obtain by passage to the colimit: 452 Chapter 18. Duality (18.5.1) Lemma. The DKL yield a natural transformation, i.e., the diagram Lhk.K; L/ DKL hnk.M X L; M X K/ Lhk.K0; L0/ DK0L0 hnk.M X L0; M X K0/ is commutative for each inclusion.K0; L0/.K; L/. (18.5.2) Lemma. The sequence! Lhk.K/ Lhk.L/ Lı LhkC1.K; L/! is exact. Similarly for triples of compact subsets. Proof. This is a special case of the general fact that a colimit over a directed set of exact sequences is again exact. A direct veriﬁcation from the deﬁnitions and the exact sequences for the representing elements is not difﬁcult. An example should sufﬁce. Suppose Lı.x/ D 0 and let x be represented by x1 2 hk.V /. We use the representing element ı.x1/ 2 hkC1.U; V / for Lı.x/. Since Lı.x/ is zero, x1 is contained in the kernel of some restriction hkC1.U; V /! hkC1.U 0; V 0/. Another representative of x is the restriction x2 2 hk.V 0/ of x1. By exactness, x2 has a pre-image in hk.U 0/, and it represents a pre-image of x in Lhk.K/. (18.5.3) Lemma. Each compact pair K; L is exc
isive for Lhk, i.e., Lhk.K [ L; K/! Lhk.L; K \ L/ is an isomorphism. Proof. This is a consequence of the isomorphisms hk.U [ V; U / Š hk.V; U \ V / for open neighbourhoods U K, V L. (18.5.4) Corollary. For each compact pair there exist an exact MV-sequence! Lhk.K [ L/! Lhk.K/ ˚ Lhk.L/! Lhk.K \ L/ ı! : Proof. The MV-sequence is constructed by algebra from suitable exact sequences using (18.5.2) and (18.5.3). (18.5.5) Lemma. For each pair.K; L/ the diagram Lhk.L/ DL hnk.M; M X L/ Lı LhkC1.K; L/ DKL @ hnk1.M X L; M X K/ is commutative. 18.5. Proof of the Duality Theorem 453 Proof. By passage to the colimit this is a consequence of the commutativity of the diagrams ı hk1.V / DV ; L; hk.U; V / DU V KL hnkC1.M; M X L/ @ hnk.M X L; M X K/: In order to verify this commutativity we apply (18.2.3) to the sets.U; V; U 0; V 0/ D.V; ;; U X L; U X K/: The element arises from oK via the excision hn.U; U X K/ Š hn.M; M X K/. One now veriﬁes from the deﬁnitions that the element ˛ is zV ; L; and ˇ becomes zU V KL. These data yield the commutative diagram hk1.V / D hk1.V / Z zV ; L; ı hk.U; V / hk1.V X L/ ı hk.U X L; V X L/ Z zU V KL hnkC1.V; V X L/ @ hnk.
V X L; V X K/ hnk.U X L; U X K/ hnkC1.M; M X L/ @ hnk.M X L; M X K/: The upper and lower rectangles commute by naturality of @ and ı. For the proof of the duality theorem we note that it sufﬁces to consider the special case DK W Lhk.K/! hnk.M; M XK/, by the Five Lemma and the previous results. The proof is based on the following principle. (18.5.6) Theorem. Let D.K/ be an assertion about compact subsets in M. Suppose: (1) D.K/ holds for sets K in a chart domain which are mapped onto a convex subset of Rn under the coordinate map. (2) If D.K/; D.L/, and D.K \ L/ hold, then also D.K [ L/ holds. T (3) Let K1 K2, K D Ki. If D.Ki / holds for each i, then D.K/ holds. Under these assumptions, D.K/ holds for all compact K. 454 Chapter 18. Duality Proof. Since an intersection of convex sets is convex, (1) and (2) yield by induction on t that D.K1 [ [ Kt / holds for compact subsets Ki of type (1) which are contained in the same chart domain. If K is a compact set in a chart domain, then K is the intersection of a sequence K1 K2 where each Ki is a ﬁnite union of compact convex sets. Each compact set K is a ﬁnite union of compact sets in chart domains. Again D.K/ follows by induction from (2). We now verify (1)–(3) of (18.5.6) in the case that D.K/ is the assertion: DK is an isomorphism. (2) The duality homomorphisms D yield a morphism of the MV-sequence for K; L into the MV-sequence of the complements; this follows from the fact that the (co-)boundary operators of the MV-sequences are deﬁned from induced morphisms and ordinary (co-)boundary operators. Now use the Five Lemma. (1)
Let'W U! Rn be a chart, K U and '.K/ convex. We begin with the special case of a point K and '.K/ D f0g. We have a commutative diagram hk.Rn/'Z'.zU K / hnk.Rn; Rn X 0/ Š hk.U / ZzU K Lhk.K/ DK hnk.U; U X K/'Š hnk.M; M X K/: The right square commutes by deﬁnition of DK; here zU K is the image of the orientation under the restriction hn.M; M X K/! hn.U; U X K/. The left square commutes by naturality of the cap product. By deﬁnition of the orientation, '.zU K / D ˙en. The fact that Zen is an isomorphism follows from the compatibility with suspension and the unit element axiom of the pairing. Hence D.K/ holds for a point K. Let now K be arbitrary and P K a point. From naturality we see that D.K/ holds, if Lhk.K/! Lhk.P / and hnk.M; M X K/! hnk.M; M X P / are isomorphisms. The set X D '.K/, being compact convex, is the intersection of a sequence of open neighbourhoods U1 U2 which are contractible onto P, and each neighbourhood of X contains eventually all Uj. Hence the restriction hk.Uj /! hk.Uj C1/ are isomorphisms, and we see hk.Uj / Š Lhk.X/ Š hk.X/. This shows the ﬁrst isomorphism. The second isomorphism is veriﬁed by standard methods (excision, h-equivalence). (3) We show that the canonical maps colimi Lhk.Ki /! Lhk.K/; colimi hnk.M; M X Ki /! hnk.M; M X K/ are isomorphisms. 18.6. Manifolds with Boundary 455 The ﬁrst isomorphism is an immediate consequence of the colim-deﬁ
nition. Given x 2 Lhk.K/ represented by y 2 hk.U /. There exists i such that Ki U. Hence y represents an element in Lhk.Ki /. This shows surjectivity, and injectivity is shown by a similar argument. The second isomorphism is easily seen for singular homology, if one uses that singular chains have compact carrier. In the general case one uses that additive homology theories commute with colimits. 18.6 Manifolds with Boundary We now treat duality for manifolds with boundary. (18.6.1) Theorem. Let M be a compact n-manifold with boundary @M, oriented by a fundamental class ŒM 2 Hn.M; @M I Z/. Then H p.M I G/! Hnp.M; @M I G/; x 7! x Z ŒM ; x 7! x Z ŒM ; H p.M; @M I G/! Hnp.M I G/; are isomorphisms for each coefﬁcient group G. Proof. By naturality and stability of the cap product the following diagram commutes up to sign (coefﬁcients are G); Œ@M D @ŒM is a fundamental class: H p.M; @M / j H p.M / i H p.@M / ı H pC1.M; @M / Z ŒM Z ŒM Z Œ@M Z ŒM Hnp.M / j Hnp.M; @M / @ Hnp1.@M / i Hnp1.M /. We know already that Z Œ@M is an isomorphism. Therefore it sufﬁces to show that the left-most vertical map is an isomorphism. We reduce the problem to the duality already proved. We use the non-compact auxiliary manifold P D M [.@M Œ0; 1Œ/; which is obtained by the identiﬁcation x.x; 0/ for x 2 @M. We also use the subspaces M.t/ D M [.@M Œ0; t/; P.t/ D M [.@M Œ0; t Œ/; 0 t < 1; 0 < t 1
: The P.t/ are a coﬁnal system of open neighbourhoods of M D M.0/ in P. The M.t/ are a compact exhaustion of P. The inclusions M P.t/ M.t / are 456 Chapter 18. Duality h-equivalences. Let i.t/ W.M; @M /.P.t/; P.t/ X M ı/. The diagram H p.M / ZŒM i.t/ H p.P.t// Zi.t/ŒM Hnp.M; @M / i.t/ Hnp.P.t/; P.t/ X M ı/ is commutative (naturality of the cap product). The map i.t/ is an isomorphism, since i.t/ is an h-equivalence; the map i.t/ is an excision. Thus it sufﬁces to show that Zi.t/ŒM is an isomorphism. We use the duality theorem for P. We have isomorphisms Hn.P; P X M.t// Š Hn.P; P X M / Š Hn.P; P X M ı/ Š Hn.M; @M /: Let z.t/ 2 Hn.P; P X M.t// correspond to the fundamental class ŒM. One veriﬁes that z.t/ is an orientation along M.t /. Since the M.t / form a compact exhaustion, the coherent family of the z.t/ yields an orientation of P. Let w.t/ 2 Hn.P.t/; P.t/ X M / and v.t/ 2 Hn.P.t/; P.t/ X M ı/ correspond to the fundamental class under Hn.P.t/; P.t/ X M / Š Hn.P.t/; P.t/ X M ı/ Š Hn.M; @M /: By deﬁnition of the duality homomorphism, DM W is the colimit of the maps LH p.M /! Hnp.P; P X M / H p.P.t// Zw.t/ Hnp.P.t/
; P.t/ X M / Š Hnp.P; P X M /: Since H p.P.t//! H p.M / is an isomorphism, the canonical maps H p.P.t//! LH p.M /! H p.M / are isomorphisms. Since DM is an isomorphism, so is Zw.t/. The diagram i H p.P.t// Zw.t/ H p.P.t// Zv.t/ Hnp.P.t/; P.t/ X M / i Hnp.P.t/; P.t/ X M ı/ is commutative by naturality of the cap product and v.t/ D iw.t/. Since v.t/ D i.t/ŒM, the map Zi.t/ŒM is an isomorphism. (18.6.2) Proposition. Let B be a compact.n C 1/-manifold with boundary @B D M. Then.M / D.1 C.1/n/.B/. In particular.M / is always even. 18.7. The Intersection Form. Signature 457 Proof. Let M D B [.@B Œ0; 1Œ/. Then B is a compact deformation retract of M and M X B Š @B0; 1Œ' @B. Hence.B/ D.M / D.M X B/ C.1/nC1.B/ D.@B/.1/n.B/. (18.6.3) Example. RP 2n is not the boundary of a compact manifold, since.RP 2n/ is odd. The same holds for an arbitrary ﬁnite product of even-dimensional real projective spaces. Problems 1. Suppose @M D A C B is a decomposition into two closed submanifolds. The diagram H p.M / H p.M; A/ H p.A/ ZŒM Hnp.M; B/ ZŒM Hnp.M; @M / ZŒ@M Hnp1.@M; A/ is commutative up to sign. Hence ZŒM W H p.M; A/! Hnp.M; B/ is an isomorphism. 18.7 The Intersection Form.
Signature Let M be a closed n-manifold oriented by a fundamental class ŒM 2 Hn.M I K/, coefﬁcients in a ﬁeld K. The evaluation on the fundamental class is H n.M /! K; x 7! xŒM D hx; ŒM i. We can also write this as the composition H n.M / Š Hom.Hn.M /; K/! K where we evaluate a homomorphism on ŒM. The canonical map " W H0.M /! K allows us to write hx; ŒM i D".x \ ŒM /. (18.7.1) Proposition. The bilinear form H k.M / H nk.M / Y H n.M / h ;ŒM i K is regular. We write this form also as.x; y/ 7! x ˇ y. Proof. We use the rule.x Yy/ZŒM D x Z.y ZŒM /. It gives us the commutative diagram H k.M / H nk Š id ZŒM H k.M / Hk.M / Š ˛id Y Z Hom.Hk.M /; K/ Hk.M / eval H n.M / ZŒM H0.M / " K. The bilinear form in question is isomorphic to the Hom-evaluation, and the latter is for each ﬁnite-dimensional vector space a regular form. 458 Chapter 18. Duality We take now R D K as coefﬁcients and assume n D 4t. In that case H 2t.M / H 2t.M /! R;.x; y/ 7! x ˇ y is a regular symmetric bilinear form. Recall from linear algebra: Let.V; ˇ/ be a real vector space together with a symmetric bilinear form ˇ. Then V has a decomposition V D VC ˚ V C V0 such that the form is positive deﬁnite on VC, negative deﬁnite on V and zero on V0. By Sylvester’s theorem the dimensions of VC and V are determined by ˇ. The integer dim VC dim V
is called the signature of ˇ. We apply this to the intersection form and call.M / D dim H 2t.M /C dim H 2t.M / the signature of the closed oriented 4t-manifold M. We also set.M / D 0, if the dimension of M is not divisible by 4. If M denotes the manifold with the opposite orientation, then one has.M / D.M /. If M D M1 C M2 then H 2t.M / D H 2t.M1/ C H 2t.M2/, the forms on M1 and M2 are orthogonal, hence.M1 C M2/ D.M1/ C.M2/. (18.7.2) Proposition. The signature of CP 2n with its natural orientation induced by the complex structure is 1. Proof. Since H 2n.CP 2nI Z// is the free abelian group generated by cn, the claim follows from hcn; ŒCP ni D1. For n D 1 this holds by the deﬁnition of the ﬁrst Chern class (see (19.1.2)). Consider the map p W.CP 1/n! CP n that sends P n iD0 cj xj ynj..Œa1; b1; : : : ; Œanbn/ to Œc0; : : : ; cn where Note that H 2..CP 1/nI Z/ is the free abelian group with basis t1; : : : ; tn where tj is the ﬁrst Chern class of pr j./. One veriﬁes that p.c/ D t1 C C tn. This implies pcn D nŠ t1t2 : : : tn. The map p has degree nŠ. These facts imply n j D1.aj x C bj y/ D Q nŠ D hpcn; ŒCP 1n i D hcn; pŒCP 1n i D hcn; nŠ ŒCP ni DnŠ hcn; ŒCP ni; hence hcn; ŒCP ni D1. (18.7.3) Proposition. Let M and N be closed oriented manifolds. Give M N the product orientation. Then.M
N / D.M /.N /. Proof. Let m D dim M and n D dim M. If m C n 6 mod4 then.M N / D 0 D.M /.N / by deﬁnition. In the case that m C n D 4p we use the Künneth isomorphism and consider the decomposition H 2p.M N / D H m=2.M / ˝ H n=2.N / L 2i<m H i.M / ˝ H 2pi.N / ˚ H mi.M / ˝ H n2pCi.N / : The ﬁrst summand on the right-hand side is zero if m or n is odd. The form on H 2p.M N / is transformed via the Künneth isomorphism by the formula 18.7. The Intersection Form. Signature 459.x ˝ y/ ˇ.x0 ˝ y0/ D.1/jyjjx0j.x ˇ x0/.y ˇ y0/. Products of elements in different summands never contribute to the top dimension m C n. Therefore the signature to be computed is the sum of the signatures of the forms on the summands. Consider the ﬁrst summand. If m=2 and n=2 are odd, then the form is zero. In the other case let A be a basis of H m=2.M / such that the form has a diagonal matrix with respect to this basis and let B be a basis of H n=2.N / with a similar property. Then.a ˝ b j a 2 A; b 2 B/ is a basis of H m=2.M / ˝ H n=2.N / for which the form has a diagonal matrix. Then.M /.N / D D a2A a ˇ a b2B b ˇ b.a;b/2AB.a ˝ b/ ˇ.a ˝ b/ D.M N /: P P P Now consider the summand for 2i < m. Choose bases A of H i.M / and B of H 2pi.N / and let A, B be the dual bases of H mi.M /, H n2p
Ci.N / respectively. Then.a ˝ b C a ˝ b; b 2 B/ is a basis of the summand under consideration. The product of different basis elements is zero, and.a ˝ b C a ˝ b/2 D.a ˝ b a ˝ b/2 shows that the number of positive squares equals the number of negative squares. Hence these summands do not contribute to the signature. There exists a version of the intersection form for cohomology with integral coefﬁcients. We begin again with the bilinear form s W H k.M /! H nk.M /! Z;.x; y/ 7!.x Y y/ŒM : We denote by A} the quotient of the abelian group A by the subgroup of elements of ﬁnite order. We obtain an induced bilinear form s} W H k.M /} H nk.M /}! Z: (18.7.4) Proposition. The form s} is regular, i.e., the adjoint homomorphism H k.M /}! Hom.H nk.M /}; Z/ is an isomorphism (and not just injective). Proof. We use the fact that the evaluation H k.M I Z/} Hk.M I Z/}! Z is a regular bilinear form over Z. By the universal coefﬁcient formula, the kernel of H k.M I Z/! Hom.Hk.M I Z/; Z/ is a ﬁnite abelian group; hence we have isomorphisms H k.M I Z/} Š Hom.Hk.M I Z/; Z/ Š Hom.Hk.M I Z/}; Z/: Now the proof is ﬁnished as before. 460 Chapter 18. Duality We return to ﬁeld coefﬁcients. Let M be the oriented boundary of the compact oriented manifold B. We set Ak D Im.i W H k.B/! H k.M // with the inclusion i W M B. (18.7.5) Proposition. The kernel of H k.M / Š! Hom.H nk.M /; K/! Hom.Ank; K/
is Ak. The isomorphism is x 7!.y 7! hy Yx; ŒM i/; the second map is the restriction to Ank. In particular dim H k.M / D dim AkCdim Ank, and in the case n D 2t we have dim H t.M / D 2 dim Ak and dim Ht.M / D 2 dim Ker.i W Ht.M /! Ht.B//. Proof. Consider the diagram H k.B/ i H k.M / ı H kC1.B; M / Š Z ŒM Š Z ŒB Hnk.M / i Hnk.M /. By stability of the cap product, the square commutes up to the sign.1/k. By commutativity and duality x 2 Ak, ı.x/ D 0, ı.x/ Z ŒB D 0, i.x Z ŒM / D 0: The regularity of the pairing H j.M / Hj.M /! K,.x; y/ 7! hx; y i says that i.x Z ŒM / D 0 is equivalent to hH nk.B/; i.x Z ŒM /i D0. Properties of pairings yield hH nk.B/; i.x Z ŒM i D hi H nk.B/; x Z ŒM i D hAnk; x Z ŒM i D hAnk Y x; ŒM i and we see that x 2 Ak is equivalent to hAnk Y x; ŒM i D0, and the latter describes the kernel of the map in the proposition. (18.7.6) Example. If n D 2t and dim H t.M / is odd, then M is not a boundary of a K-orientable compact manifold. This can be applied to RP 2n (for K D Z=2) and to CP 2n (for K D R). Þ (18.7.7) Proposition. Let M be the boundary of a compact oriented.4k C 1/manifold B. Then the signature of M is zero. 18.8. The Euler Number 461 Proof. It follows from Proposition (18.7.5) that the orthogonal complement of A2k with respect to
the intersection form on H 2k.M I R/ is A2k, and 2 dim A2k D dim H 2k.M I R/. Linear algebra tells us that a symmetric bilinear form with these properties has signature 0. We generalize the preceding by taking advantage of the general duality isomorphism (coefﬁcients in K). Let M be an n-manifold and K L a compact pair in M. Assume that M is K-oriented along K. We deﬁne a bilinear form./ LH i.K; L/ H j.M X L; M X K/ Y! H iCj.M; M X K/ as follows: Let.V; W / be a neighbourhood of.K; L/. We ﬁx an element y 2 H j.M X L; M X K/ and restrict it to H j.V X L; V X K/. Then we have H i.V; W /! H i.V X L; W X L/ H iCj.M; W [.M X K//! H iCj.M; M X K/: Yy! H iCj.V X L; W X K [ V X K/ Š The colimit over the neighbourhoods.V; W / yields Yy in./. (18.7.8) Proposition. Let M be an n-manifold and K L compact ENR in M. Then H i.K; L/ H ni.M X L; M X K/ Y H n.M; M X K/ h ;oK i K is a regular bilinear form. (18.7.9) Example. Let M be a compact oriented n-manifold for n 2 mod.4/. Then the Euler characteristic.M / is even. The intersection form H n=2.M / H n=2.M /! Q with coefﬁcients in Q is skew-symmetric and regular, since n=2 is odd. By linear algebra, a form of this type only exists on even-dimensional vector spaces..M / D P n iD0.1/i dim Hi.M / P D dim H n=2.M / C 2 2i<n.1/i dim H i.M /I we have used dim
H i.M / D dim H ni.M /, and this holds because of H i.M / Š Hom.Hi.M /; Q/ and hence dim H i.M / D dim Hi.M / D dim H ni.M /. Þ 18.8 The Euler Number Let W E./! M be an n-dimensional real vector bundle over the closed connected orientable manifold M. The manifold is oriented by a fundamental class ŒM 2 Hn.M I Z/ and the bundle by a Thom class t./ 2 H n.E; E0I Z/. Let s W M! 462 Chapter 18. Duality E./ be a section of and assume that the zero set N.s/ is contained in the disjoint sum D D D1 [ [ Dr of disks Dj. The aim of this section is to determine the Euler number e./ D hst./; ŒM i by local data. We assume given positive charts 'j W Rn! Uj with disjoint images of M such that 'j.Dn/ D Dj. The bundle is trivial over Uj. Let Dn Rn pr Dn ˆj 'j E.jDj / Dj be a trivialization. We assume that ˆj is positive with respect to the given orientation of. These data yield a commutative diagram H n.E./; E0.// s H n.M / ˛ H n.E.jD/ [ E 0.jM X Dı; E0.jD// s H n.; M X Dı/ H n.E.jD/; E0.jD// L Š j H n.E.jDj /; E0.jDj // s s j ˇ Š H n.D; S/ Š L j H n.Dj ; Sj /. Here Sj is the boundary of Dj and S D j Sj. The restriction of s to Dj is sj. The image of t./ in H n.E.jDj /; E0.jDj // is the Thom class t.jDj /. The vertical maps have their counterpart in homology S Hn.M / ˛ Hn.M; M X Dı/ ˇ Š Hn.D; S/ Š L j
Hn.Dj ; Sj /; and ŒM is mapped to.ŒDj /. By commutativity and naturality hst./; ŒM i D P j hs j t.jDj /; ŒDj i: The bundle isomorphism.ˆj ; 'j / transports sj into a section tj W Dn! Dn Rn; x 7!.x; uj.x// of pr. Note that uj.S n1/ Rn X 0. We have another commutative diagram 18.8. The Euler Number 463 H n.E.jDj /; E0.jDj // ˆ j H n.Dj ; Sj /'j H n.Dn Rn; Dn Rn 0/ H n.Dn; S n1/ s j t j pr u j H n.Rn; Rn 0/. The evaluation hs 0 if we choose the correct orientations. We explain this now and use the following computation: j t.jDj /; ŒDj i is the degree of uj W S n1! Rn hs j t.jDj /; ŒDj i D hs D ht D d.uj /hen; en i: j s j t.jDj /; 'j i D h' j t.jDj /; en i D hu j ˆ j t.jDj /; en i j en; en i The cohomological degree of uj is u obtain: j en D d.uj /en. With these deﬁnitions we (18.8.1) Proposition. e./ D P r j D1 d.uj / hen; en i. If s has in Dj an isolated zero, then d.uj / is called the index of this zero. Problems 1. There always exists a section with a single zero. 2. The index can be computed for transverse zeros of a smooth section s W M! E of a smooth bundle. Consider the differential Txs W TxM! TxE D TxM ˚ Ex: Transversality means that the composition with the projection pr ıTxs W TxM! Ex is an isomorphism between oriented vector spaces. This isomorphism has a
sign ".x/ 2 f˙1g, C1 if the orientation is preserved. Show that ".x/ is the local index. 3. The section s W S n! TS n S n RnC1; x D.x0; : : : ; xn/ 7!.x;.x2 0 1; x0x1; : : : ; x0xn// has the transverse zeros.1; 0; : : : ; 0/ with index 1 and.1; 0; : : : ; 0/ with index.1/n. 4. Find a vector ﬁeld on S 2n with a single zero (of index 2). 5. There exists a section without zeros if and only if the Euler number is zero. We know already that the Euler number is zero, if there exists a non-vanishing section. For the converse one has to use two facts: (1) There always exist sections with isolated zeros. 464 Chapter 18. Duality (2) There exists a cell D which contains every zero. Hence one has to consider a single local index. This index is zero, and the corresponding map u W S! Rn 0 is null homotopic. Thus there exists an extension u W D! Rn 0. We use this extension to extend the section s over the interior of D without zeros. 6. Consider the bundle.k/ W H.k/! CP 1. Let P.z0; z1/ D homogeneous polynomial of degree k. Then j D0 ˛kzj 0 zkj be a P k 1 W CP 1! H.k/; Œz0; z1 7!.z0; z1I P.z0; z1// Q j.aj z1 bj z0/ is the factorization into linear factors, is a section of.k/. If P.z0; z1/ D then the Œaj ; bj 2 CP 1 are the zeros of, with multiplicities. 7. Consider the bundle W S n Z=2 Rn! RP n,.x; z/ 7! Œx. Then W Œx0; : : : ; xn 7!..x0; : : : ; xn/;.x1; : : : ; xn// is a section with a single

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