On the construction and topological invariance of the Pontryagin classes
aa r X i v : . [ m a t h . A T ] F e b ON THE CONSTRUCTION AND TOPOLOGICAL INVARIANCE OFTHE PONTRYAGIN CLASSES
ANDREW RANICKI AND MICHAEL WEISS
Abstract.
We use sheaves and algebraic L -theory to construct the rational Pontryaginclasses of fiber bundles with fiber R n . This amounts to an alternative proof of Novikov’stheorem on the topological invariance of the rational Pontryagin classes of vector bundles.Transversality arguments and torus tricks are avoided. Introduction
The “topological invariance of the rational Pontryagin classes” was originally the state-ment that for a homeomorphism of smooth manifolds, f : N → N ′ , the induced map f ∗ : H ∗ ( N ′ ; Q ) → H ∗ ( N ; Q )takes the Pontryagin classes p i ( T N ′ ) ∈ H i ( N ′ ; Q ) of the tangent bundle of N ′ to thePontryagin classes p i ( T N ) of the tangent bundle of N . The topological invariance wasproved by Novikov [Nov], some 40 years ago. This breakthrough result and its torus-relatedmethod of proof have stimulated many subsequent developments in topological manifolds,notably the formulation of the Novikov conjecture, the Kirby-Siebenmann structure theoryand the Chapman-Ferry-Quinn et al. controlled topology.The topological invariance of the rational Pontrjagin classes has been reproved severaltimes (Gromov [Gr], Ranicki [Ra3], Ranicki and Yamasaki [RaYa]). In this paper we giveyet another proof, using sheaf-theoretic ideas. We associate to a topological manifold M a kind of “tautological” co-sheaf on M of symmetric Poincar´e chain complexes, in such away that the cobordism class is a topological invariant by construction. We then produceexcision and homotopy invariance theorems for the cobordism groups of such co-sheaves,and use the Hirzebruch signature theorem to extract the rational Pontryagin classes of M from the cobordism class of the tautological co-sheaf on M . Definition 1.1.
Write TOP( n ) for the space of homeomorphisms from R n to R n and PL( n )for the space (geometric realization of a simplicial set) of PL-homeomorphisms from R n to R n . Let TOP = S n TOP( n ) and PL = S n PL( n ).The topological invariance can also be formulated in terms of the classifying spaces,as the statement that the homomorphism H ∗ ( B TOP; Q ) → H ∗ ( B O; Q ) induced by theinclusion O → TOP is onto. Here H ∗ ( B O; Q ) = Q [ p , p , p , . . . ]where p i ∈ H i ( B O; Q ) is the i -th “universal” rational Pontryagin class. We note also thatthe inclusion B SO → B O induces an isomorphism in rational cohomology, and the inclusion
Date : January 7, 2009. The point is that topological n -manifolds have topological tangent bundles with structure group TOP( n )which are sufficiently natural under homeomorphisms. See [Kis]. B STOP → B TOP induces a surjection in rational cohomology by a simple transfer argu-ment. Therefore it is enough to establish surjectivity of H ∗ ( B STOP; Q ) → H ∗ ( B SO; Q ).Hirzebruch’s signature theorem [Hirz] expressed the signature of a closed smooth ori-ented 4 i -dimensional manifold M as the evaluation on the fundamental class [ M ] ∈ H i ( M )of the L -genus L ( T M ) ∈ H ∗ ( M ; Q ) , that is signature( M ) = hL ( T M ) , [ M ] i ∈ Z ⊂ Q . It follows that, for any closed smooth oriented n -dimensional manifold N and a closed 4 i -dimensional framed submanifold M ⊂ N × R k (where framed refers to a trivialized normalbundle), signature( M ) = hL ( T N ) , [ M ] i ∈ Z ⊂ Q . By Serre’s finiteness theorem for homotopy groups, H n − i ( N ; Q ) ∼ = lim −→ k [Σ k N + , S n − i + k ] ⊗ Q with N + = N ∪ { pt. } , and by Pontryagin-Thom theory [Σ k N + , S n − i + k ] can be identifiedwith the bordism group of closed framed 4 i -dimensional submanifolds M ⊂ N × R k . It isthus possible to identify the component of L ( T N ) in H i ( N ; Q ) ∼ = hom( H n − i ( N ; Q ) , Q )with the linear map H n − i ( N ; Q ) −→ Q ; M signature( M ) . The assumption that N be closed can be discarded if we use cohomology with compactsupports where appropriate. Then we identify the component of L ( T N ) in H i ( N ; Q ) ∼ = hom( H n − ic ( N ; Q ) , Q )with the linear map H n − ic ( N ; Q ) → Q ; M signature( M ) . Now we can choose N in such a way that the classifying map κ : N → B SO( n ) for the tangentbundle is highly connected, say (4 i +1)-connected. Then κ ∗ : H i ( B SO( n ); Q ) → H i ( N ; Q )takes L of the universal oriented n -dimensional vector bundle to L ( T N ), and so the abovedescription of L ( T N ) in terms of signatures can be taken as a definition of the universal
L ∈ H i ( B SO( n ); Q ), or even L ∈ H i ( B SO; Q ).Since this definition relies almost exclusively on transversality arguments, which carryover to the PL setting, we can deduce immediately, as Thom did, that Hirzebruch’s L -genusextends to L ∈ H ∗ ( B SPL; Q ) . As the rational Pontryagin classes are polynomials (with rational coefficients) in the com-ponents of L , the PL invariance of the rational Pontryagin classes follows from this ratherstraightforward argument. It was also clear that the topological invariance of the rationalPontryagin classes would follow from an appropriate transversality theorem in the settingof topological manifolds. However, Novikov’s proof did not exactly deduce the topologicalinvariance of the Pontryagin classes from a topological transversality statement. Instead,he proved that signatures of the submanifolds were homeomorphism invariants by showingthat for a homeomorphism f : N → N ′ of closed smooth oriented n -dimensional manifolds N, N ′ and a closed framed 4 i -dimensional submanifold M ′ ⊂ N ′ × R k it is possible to make ONTRYAGIN CLASSES 3 the proper map f × id : N × R k → N ′ × R k transverse regular at M ′ , keeping it proper, andthe smooth transverse image M ⊂ N × R k hassignature( M ) = signature( M ′ ) ∈ Z . This was done using non-simply-connected methods. Subsequently, it was found that theideas in Novikov’s proof could be extended and combined with non-simply-connected surgerytheory to prove transversality for topological (non-smooth, non-PL) manifolds. Details onthat can be found in [KiSi]. It is now known that H ∗ ( B TOP; Q ) ∼ = H ∗ ( B O; Q ).The collection of the signatures of framed 4 i -dimensional submanifolds M ⊂ N × R k ofa topological n -dimensional manifold N was generalized in Ranicki [Ra2] to a fundamentalclass [ N ] L ∈ H n ( N ; L • ) with coefficients in a spectrum L • of symmetric forms over Z .However, [Ra2] made some use of topological transversality. This paper will remedy thisby expressing [ N ] L as the cobordism class of the tautological co-sheaf on N , using the localPoincar´e duality properties of N .In any case the “modern” point of view in the matter of the topological invarianceof rational Pontryagin classes is that it merits a treatment separate from transversalitydiscussions. Topological manifolds ought to have (tangential) rational Pontryagin classesbecause they satisfy a local form of Poincar´e duality. More precisely, if N is a topological n -manifold, then for any open set U ⊂ N we have a Poincar´e duality isomorphism betweenthe homology of U and the cohomology of U with compact supports. The task is then touse this refined form of Poincar´e duality to make invariants in the homology or cohomologyof N . Whether or not these invariants can be expressed as characteristic classes of thetopological tangent bundle of N becomes a question of minor importance. Remark.
In this paper we make heavy use of homotopy direct and homotopy inverselimits of diagrams of chain complexes and chain maps. They can be defined like homotopydirect and homotopy inverse limits of diagrams of spaces. That is to say, the Bousfield-Kanformulae [BK] for homotopy direct and homotopy inverse limits of diagrams of spaces caneasily be adapted to diagrams of chain complexes: products (of spaces) should be replacedby tensor products (of chain complexes), and where standard simplices appear they should,as a rule, be replaced by their cellular chain complexes.We often rely on [DwK, § Duality and L -theory: Generalities In the easiest setting, we start with an additive category A and the category C of allchain complexes in it, graded over Z and bounded from above and below. We assume givena functor ( C, D ) C ⊠ D from C ×C to chain complexes of abelian groups. This is subject to bilinearity and symmetryconditions: • for fixed D in C , the functor C C ⊠ D takes contractible objects to contractible ob-jects, and takes homotopy cocartesian (= homotopy pushout) squares to homotopycocartesian squares ; • there is a binatural isomorphism τ : C ⊠ D → D ⊠ C satisfying τ = id.We assume that every object C in C has a “dual” (w.r.t. ⊠ ). This means that the functor D H ( C ⊠ D ) ANDREW RANICKI AND MICHAEL WEISS is co-representable in the homotopy category HC , so that there exists C −∗ in C and a naturalisomorphism H ( C ⊠ D ) ∼ = [ C −∗ , D ]where the square brackets denote chain homotopy classes of maps. Let’s note that in thiscase the n -fold suspension Σ n C −∗ co-represents the functor C H n ( C ⊠ D ) in HC . Example 2.1. A is the category of f.g. free left modules over Z [ π ], for a fixed group π .For C and D in C we let C ⊠ D = C t ⊗ Z [ π ] D , using the standard involution P a g g P a g g − on Z [ π ] to turn C into a chain complex ofright Z [ π ]-modules C t . Then the dual C −∗ of any C in C exists and can be defined explicitlyas the chain complex of right module homomorphisms hom Z [ π ] ( C t , Z [ π ]) on which Z π actsby left multiplication: ( rf )( c ) := r ( f ( c ))for r ∈ Z [ π ], c ∈ C t and f ∈ hom Z [ π ] ( C t , Z [ π ]). Definition 2.2. An n -dimensional symmetric algebraic Poincar´e complex in C consists ofan object C in C and an n -dimensional cycle ϕ in ( C ⊠ C ) h Z / whose image in H n ( C ⊠ C )is nondegenerate (i.e., adjoint to a homotopy equivalence Σ n C −∗ → C ). Remark . ( C ⊠ C ) h Z / = hom Z [ Z / ( W, C ⊠ C ) where W is your favorite projectiveresolution of the trivial module Z over the ring Z [ Z / Example : Suppose that A is the category of f.g. free abelian groups. Let C be thecellular chain complex of a space X which is the realization of a f.g. simplicial set, andalso a Poincar´e duality space of formal dimension n . Then you can use an Eilenberg-Zilberdiagonal W ⊗ C −→ C ⊗ C (respecting Z / C → ( C ⊗ C ) h Z / on a fundamental cycleto get a nondegenerate n -cycle ϕ ∈ ( C ⊗ C ) h Z / . Definition 2.4.
An ( n + 1)-dimensional symmetric algebraic Poincar´e pair in C consistsof a morphism f : C → D in C , an n -dimensional cycle ϕ in ( C ⊠ C ) h Z / and an ( n + 1)-dimensional chain ψ in ( D ⊠ D ) h Z / such that ( f ⊠ f )( ϕ ) = ∂ψ and • the image of ϕ in H n ( C ⊠ C ) is nondegenerate, • the image of ψ in H n +1 ( D ⊠ D/C ) is nondegenerate, where
D/C is shorthand forthe algebraic mapping cone of f : C → D .The boundary of the SAP pair ( C → D, ψ, ϕ ) is (
C, ϕ ), an n -dimensional SAPC.With these definitions, it is straightforward to design bordism groups L n ( C ) of n -dimensional SAPCs in C . Elements of L n ( C ) are represented by n -dimensional SAPCs in C .We say that two such representatives, ( C, ϕ ) and ( C ′ , ϕ ′ ), are bordant if ( C ⊕ C ′ , ϕ ⊕ − ϕ ′ )is the boundary of an SAP pair of formal dimension n + 1.Furthermore there exist generalizations of the definitions of SAPC and SAP pair (going inthe direction of m -ads) which lead automatically to the construction of a spectrum L • ( C )such that π n L • ( C ) ∼ = L n ( C ). These activities come under the heading symmetric L -theoryof C . This is an idea going back to Mishchenko [Mis]. It was pointed out in [Ra1] that thereis an analogue of Mishchenko’s setup, the quadratic L -theory of C , where “homotopy fixedpoints” of Z / ONTRYAGIN CLASSES 5
Definition 2.5. An n -dimensional quadratic algebraic Poincar´e complex in C consists ofan object C in C and an n -dimensional cycle ϕ in ( C ⊠ C ) h Z / whose image in H n ( C ⊠ C )(under the transfer) is nondegenerate. Remark . ( C ⊠ C ) h Z / = W ⊗ Z [ Z / ( C ⊠ C ) where W is that resolution, as above.The bordism groups of QAPCs are denoted L n ( C ) and the corresponding spectrumis L • ( C ). There is a (norm-induced) comparison map L • ( C ) → L • ( C ). On the algebraicside, a key difference between L n ( C ) and L n ( C ) is that elements of L n ( C ) can always berepresented by “short” chain complexes (concentrated in degree k if n = 2 k , and in degrees k and k + 1 if n = 2 k + 1), while that is typically not the case for elements of L n ( C ).On the geometric side, L n ( C ) also has the immense advantage of being directly relevant todifferential topology as a surgery obstruction group, for the right choice(s) of C . But thecomparison maps L n ( C ) → L n ( C ) are always isomorphisms away from the prime 2, and sincewe are interested in rational questions, there is no need to make a very careful distinctionbetween symmetric and quadratic L -theory here. Example 2.7. If A is the category of f.g. free abelian groups, C the corresponding chaincomplex category, then L n ( C ) ∼ = Z n ≡ n ≡ Z / n ≡ n ≡ A is the category of f.d. vector spaces over Q , and C the corresponding chain complexcategory, then L n ( C ) ∼ = L n ( C ) ∼ = (cid:26) Z ⊕ ( Z / ∞ ⊕ ( Z / ∞ n ≡ n ≡ , , ... ) ∞ denotes a countably infinite direct sum. Remark . We need a mild generalization of the setup above. Again we start with an ad-ditive category A . Write B ( A ) for the category of all chain complexes of A -objects, boundedfrom below (but not necessarily from above). We suppose that a full subcategory K of B ( A )has been specified, closed under suspension, desuspension, homotopy equivalences, directsums and mapping cone constructions, so that the homotopy category HK is a triangulatedsubcategory of HB ( A ). We assume given a functor( C, D ) C ⊠ D from K × K to chain complexes of abelian groups. This is subject to the usual bilinearityand symmetry conditions: • for fixed D in K , the functor C C ⊠ D takes contractible objects to contractibleones and preserves homotopy cocartesian (= homotopy pushout) squares; • there is a binatural isomorphism τ : C ⊠ D → D ⊠ C satisfying τ = id.We assume that every object C in K has a “dual” (w.r.t. ⊠ ). This means that the functor D H ( C ⊠ D ) on HK is co-representable. From these data we construct L -theory spectra L • ( K ) and L • ( K ) as before. (Some forward “hints”: Our choice of additive category A isfixed from definition 3.1 onwards, and for K we take the category D ′ defined in section 4.) ANDREW RANICKI AND MICHAEL WEISS Chain complexes in a local setting
Let X be a locally compact, Hausdorff and separable space. Let O ( X ) be the posetof open subsets of X . We introduce an additive category A = A X whose objects arefree abelian groups (typically not finitely generated) equipped with a system of subgroupsindexed by O ( X ). Definition 3.1.
An object of A is a free abelian group F with a basis S , together withsubgroups F ( U ) ⊂ F , for U ∈ O ( X ), such that the following conditions are satisfied. • F ( ∅ ) = 0 and F ( X ) = F . • Each subgroup F ( U ) is generated by a subset of S . • For
U, V ∈ O ( X ) we have F ( U ∩ V ) = F ( U ) ∩ F ( V ).A morphism in A from F to F is a group homomorphism F → F which, for every U ∈ O ( X ), takes F ( U ) to F ( U ). Example 3.2.
For i ≥
0, the i -th chain group C i = C i ( X ) of the singular chain complex of X has a preferred structure of an object of A . The preferred basis is the set S i = S i ( X ) ofsingular i -simplices in X . For open U in X , let C i ( U ) ⊂ C i ( X ) be the subgroup generatedby the singular simplices with image contained in U . The boundary operator from C i ( X )to C i − ( X ) is an example of a morphism in A . Definition 3.3.
We write B ( A ) for the category of chain complexes in A , graded over Z and bounded from below.Next we list some conditions which we might impose on objects in B ( A ), to define asubcategory in which we can successfully do L -theory. The kind of object that we are mostinterested in is described in the following example. Example 3.4.
Take a map f : Y → X , where Y is a compact ENR (euclidean neighborhoodretract). Let C ( f ) be the object of B ( A ) defined as follows: C ( f )( X ) is the singular chaincomplex of Y , with the standard graded basis, and C ( f )( U ) ⊂ C ( f )( X ) for U ∈ O ( X ) isthe subcomplex generated by the singular simplices of Y whose image is in f − ( U ).We start by listing some of the obvious but remarkable properties which we see in thisexample 3.4. Let C be any object of B ( A ). Definition 3.5.
We say that C satisfies the sheaf type condition if, for any subset W of O ( X ), the inclusion X V ∈W C ( V ) −→ C ( [ V ∈W V )is a homotopy equivalence. We say that C satisfies finiteness condition (i) if the following holds. There exists an integer a ≥ V ⊂ V in X such that the closure of V is contained in V , theinclusion C ( V ) → C ( V ) factors up to chain homotopy through a chain complex D of f.g. free abelian groups, with D i = 0 if | i | > a .We say that C satisfies finiteness condition (ii) ifthere exists a compact K in X such that C ( U ) depends only on U ∩ K . (Then wesay that C is supported in K .) The sum sign is for an internal sum taken in the chain complex C ( X ), not an abstract direct sum. ONTRYAGIN CLASSES 7
Lemma 3.6.
Example 3.4 satisfies the sheaf type condition and the two finiteness condi-tions.Proof.
The sheaf type condition is well known from the standard proofs of excision in sin-gular homology. A crystal clear reference for this is [Do, III.7.3]. For finiteness condition(i), choose a finite simplicial complex Z and a retraction r : Z → Y (with right inverse j : Y → K ). Replacing the triangulation of Z by a finer one if necessary, one can find afinite simplicial subcomplex Z ′ of Z containing j ( f − U ) and such that r ( Z ′ ) ⊂ f − ( V ).Then the inclusion f − U → f − ( V ) factors through Z ′ . The singular chain complex of Z ′ is homotopy equivalent to the cellular chain complex of Z ′ , a chain complex of f.g. freeabelian groups which is zero in degrees < > dim( Z ). Finiteness condition(ii) is satisfied with K = f ( Y ). (cid:3) Definition 3.7.
Let
C ⊂ B ( A ) consist of the objects which satisfy the sheaf type conditionsand the two finiteness conditions.In the following proposition, we write C X and C Y etc. rather than C , to emphasize thedependence on a space such as X or Y . Definition 3.8.
A map f : X → Y induces a “pushforward” functor f ∗ : C X → C Y , definedby f ∗ C ( U ) = C ( f − ( U )) for C in C X and open U ⊂ Y .Returning to the shorter notation ( C for C X ), we spell out two elementary consequencesof the sheaf condition. Lemma 3.9.
Let C be an object of C and let W be a finite subset of O ( X ) . If W is closedunder unions (i.e., for any V, W ∈ W the union V ∪ W is in W ) then the inclusion-inducedmap C ( \ V ∈W V ) −→ holim V ∈W C ( V ) is a homotopy equivalence.Proof. Choose V , . . . , V k ∈ W such that every element of W is a union of some (at leastone) of the V i . For nonempty S ⊂ { , . . . k } let V S = S i ∈ S V i . There is an inclusionholim V ∈W C ( V ) −→ holim nonempty S ⊂{ ,...,k } C ( V S )and we show first that this is a homotopy equivalence. It is induced by a map f of posets. Onthe right-hand side, we have the poset of nonempty subsets of { , . . . , k } partially ordered byreverse inclusion, and on the left-hand side we have W itself, partially ordered by inclusion.The map f is given by S V S . Under these circumstances it is enough to show that f hasa (right) adjoint g . But this is clear: for V ∈ W let g ( V ) = { i | V i ⊂ V } .Now it remains to show that the inclusion-induced map C ( k \ i =1 V i ) −→ holim nonempty S ⊂{ ,...,k } C ( V S ) ANDREW RANICKI AND MICHAEL WEISS is a homotopy equivalence. We show this without any restrictive assumptions on V , . . . , V k .The map fits into a commutative square C ( T k − i =1 V i ∩ V k ) = C ( T ki =1 V i ) / / (cid:15) (cid:15) holim ∅6 = S ⊂{ ,...,k } C ( V S ) (cid:15) (cid:15) holim ∅6 = S ⊂{ ,...,k − } C ( V S ∩ V k ) / / holim ∅6 = S ⊂{ ,...,k − } holim ( C ( V S ) → C ( V S ∪ k ) ← C ( V k )) . The left-hand vertical arrow in the square is a homotopy equivalence by inductive assump-tion. The lower horizontal arrow is induced by homotopy equivalences C ( V S ∩ V k ) −→ holim ( C ( V S ) → C ( V S ∪ k ) ← C ( V k ))and is therefore also a homotopy equivalence. The right-hand vertical arrow is contravari-antly induced by a map of posets, ( S, T ) T where S is a nonempty subset of { , . . . , k − } and T = S or T = S ∪ k or T = k . Hence itis enough, by [DwK, 9.7], to verify that the appropriate categorical “fibers” of this map ofposets have contractible classifying spaces. For fixed T ′ , a nonempty subset of { , . . . , k } ,the appropriate fiber is the poset of all ( S, T ) as above with T ⊂ T ′ . It is easy to verify thatthe classifying space is contractible. (cid:3) Lemma 3.10.
Let C be an object of C and let W be a subset of O ( X ) . If W is closedunder intersections (i.e., for any V, W ∈ W the intersection V ∩ W is in W ) then theinclusion-induced map hocolim V ∈W C ( V ) −→ C ( [ V ∈W V ) is a homotopy equivalence.Proof. In the case where W is finite, the proof is analogous to that of lemma 3.9. The generalcase follows from the case where W is finite by an obvious direct limit argument. (cid:3) A zoo of subcategories
The category C = C X defined in section 3 should be regarded as a provisional workenvironment. It has two shortcomings. • A morphism f : C → D in C which induces homotopy equivalences C ( U ) → D ( U )for every open U ⊂ X need not be a chain homotopy equivalence in C .We can fix that rather easily, and will do so in this section, by defining free objects in C andshowing that all objects in C have free resolutions. This leads to a decomposition of C intofull subcategories C ′ and C ′′ , where C ′ contains the free objects and C ′′ contains the objectswhich we regards as weakly equivalent to • Given the decomposition of C into C ′ and C ′′ , we are able to set up a good dualitytheory either in C ′ or, less formally, in C modulo C ′′ . The resulting quadratic L -theory spectrum is still a functor of X , because A , C , C ′ , C ′′ depend on X . For thisfunctor we are able to prove homotopy invariance, but not excision. ONTRYAGIN CLASSES 9
We solve this problem not by adding further conditions to the list in definition 3.5, butinstead by defining a full subcategory D of C in terms of generators. To be more precise, D is generated by all objects which are weakly equivalent to 0 and all the examples of 3.4obtained from singular simplices f : ∆ k → X , using the processes of extension, suspensionand desuspension. The good duality theory in C ′ or C modulo C ′′ restricts to a good dualitytheory in D ′ or D modulo D ′′ , where D ′ = D ∩ C ′ and D ′′ = D ∩ C ′′ . The corresponding L -theory functor X L • ( D X ) satisfies homotopy invariance and excision.Unfortunately it is not clear that all the objects of C obtained as in example 3.4 belong to D .They do however belong to r D , the idempotent completion of D within C . We are not ableto prove excision for the functor X L • ( r D X ), but we do have a long exact “Rothenberg”sequence showing that the inclusion L • ( D X ) → L • ( r D X ) is a homotopy equivalence awayfrom the prime 2. C ′ (cid:15) (cid:15) (cid:15) (cid:15) r D ′ o o o o (cid:15) (cid:15) (cid:15) (cid:15) D ′ o o o o (cid:15) (cid:15) (cid:15) (cid:15) B ( A ) C o o o o r D o o o o D o o o o C ′′ O O O O r D ′′ o o o o O O O O D ′′ o o o o O O O O Definition 4.1.
A nonempty open subset U of X determines a functor F on O ( X ) by F ( V ) = (cid:26) Z if U ⊂ V F ( X ), this becomes an object of A which we call free on onegenerator attached to U . Any direct sum of such objects is called free . Example 4.2.
The singular chain group C i ( X ) of X , with additional structure as in ex-ample 3.2, is typically not free in the sense of 4.1. Definition 4.3.
Let C ′ ⊂ C be the full subcategory consisting of the objects which are freein every dimension. Let C ′′ ⊂ C be the full subcategory consisting of the objects C for which C ( U ) is contractible, for all U ∈ O ( X ). Definition 4.4.
A morphism f : C → D in C is a weak equivalence if its mapping conebelongs to C ′′ . Lemma 4.5.
Every morphism in C from an object of C ′ to an object of C ′′ is nullhomotopic.Proof. The nullhomotopy can be constructed by induction over skeleta, using the following“projective” property of free objects in A . Let a diagram B y f A −−−−→ B in A be given where f is strongly onto (that is, the induced map B ( U ) → B ( U ) is onto forevery U ) and A is free. Then there exists g : A → B making the diagram commutative. (cid:3) Lemma 4.6.
For every object D of C , there exists an object C of C ′ and a morphism C → D which is a weak equivalence.Proof. The morphism C → D can be constructed inductively using the fact that, for every B in A , there exists a free A in A and a morphism A → B which is strongly onto. (cid:3) The next lemma means that objects of C ′ are “cofibrant”: Lemma 4.7.
Let f : C → D and g : E → D be morphisms in C . Suppose that C is in C ′ and g is a weak equivalence. Then there exists a morphism f ♯ : C → E such that gf ♯ ishomotopic to f .Proof. By lemma 4.5, the composition of f with the inclusion of D in the mapping cone of g is nullhomotopic. Choosing a nullhomotopy and unravelling that gives f ♯ : C → E and ahomotopy from gf ♯ to f . (cid:3) Definition 4.8.
We write HC ′ for the homotopy category of C ′ . By all the above, this isequivalent to the category C / C ′′ obtained from C by making invertible all morphisms whosemapping cone belongs to C ′′ .In the following lemma, we write C X and C Y etc. instead of C to emphasize the depen-dence of C on a space such as X or Y . Lemma 4.9.
The “pushforward” functor f ∗ : C X → C Y determined by f : X → Y restricts toa functor C ′′ X → C ′′ Y . If f is an open embedding, it also restricts to a functor C ′ X → C ′ Y . (cid:3) This completes our discussion of freeness and weak equivalences in C . We now turn tothe concept of decomposability , which is related to excision. Definition 4.10.
Let D be the smallest full subcategory of C = C X with the followingproperties. • All objects of C obtained from maps ∆ k → X (where k ≥
0) by the method ofexample 3.4 belong to D . • If C → D → E is a short exact sequence in C and two of the three objects C, D, E belong to D , then the third belongs to D . • D ⊃ C ′′ , that is, all weakly contractible objects in C belong to 0.When we say that an object of C is decomposable , we mean that it belongs to D . Definition 4.11. D ′ := D ∩ C ′ and D ′′ := D ∩ C ′′ . Lemma 4.12.
Let C in C be obtained from a map Y → X as in example 3.4, where Y isa compact ENR. Then C is a retract in C of an object of D .Proof. Choose a retraction r : Z → Y where Z is a finite simplicial complex (with rightinverse j : Y → Z , say). Then f r : Z → X determines an object of C as in example 3.4,and this is clearly in D . The object of C determined by f is a retract of the object of D determined by f r . (cid:3) Lemma 4.13.
The rule X
7→ D X is a covariant functor. (cid:3) Lemma 4.14.
Let V and W be open subsets of X such that the closure of V in X iscontained in W . Given D in D X , there are C in D W and a morphism g : C → D in D X such that g U : C ( U ) → D ( U ) is a homotopy equivalence for any open U ⊂ V .Proof. If that claim is true for a particular D and all V, W as in the statement, then we saythat D has property P . It is enough to verify the following. ONTRYAGIN CLASSES 11 (i) Every object obtained from a map ∆ k → X by the method of example 3.4 hasproperty P .(ii) If a : D → E is a weak equivalence in D X and if one of D, E has property P , thenthe other has property P .(iii) If f : D → E is any morphism in D X , and both D and E have property P , then themapping cone of f has property P .The proof of (i) is straightforward using barycentric subdivisions. Also, one direction of (ii)is trivial: if D has property P , then E has property P . For the converse, suppose that E hasproperty P . For fixed V and W , choose g : F → E with F in D W such that F ( U ) → E ( U )is a homotopy equivalence for all U ⊂ V . Without loss of generality, F is free in everydimension. (Otherwise use lemma 4.6.) Then F belongs to D ′ X also. By lemma 4.7, thereexists a morphism h : F → D such that the composition ah : F → E is homotopic to g . Then h is a morphism which solves our problem.For the proof of (iii), we fix V and W and choose an open W such that V ⊂ W ⊂ W , andthe closure of V in X is contained in W while the closure of W in X is contained in W .Then we choose C in C W and C → E in C X inducing homotopy equivalences C ( U ) → E ( U )for all open U ⊂ W . We also choose B in C W and B → D inducing homotopy equivalences B ( U ) → D ( U ) for all open U ⊂ V . Without loss of generality, B is free in every dimension.Hence there exists B → C making the diagram B −−−−→ C y y D −−−−→ E commutative up to homotopy. Any choice of such a homotopy determines a map from themapping cone of B → C to the mapping cone of D → E which solves our problem. Inparticular the mapping cone of B → C belongs to C W . (cid:3) Corollary 4.15.
Let X = Y ∪ Z where Y, Z are open in X . Then for any D in D X , thereexists a morphism C → D in D X such that C is in D Y and the mapping cone of C → D isweakly equivalent to an object of D Z . (cid:3) Proof.
Choose an open neighborhood V of X r Z in X such that the closure of V (in X ) iscontained in Y . Apply lemma 4.14 with this V and W = Y . (cid:3) Duality in a local setting
Definition 5.1.
For an object C in C and open subsets U, V ⊂ X with V ⊂ U , let C ( U, V )be the chain complex C ( U ) /C ( V ) (of free abelian groups). Definition 5.2.
For objects C and D of C , let C ⊠ D = holim U ⊂ X open , K , K ⊂ X closed K ∩ K ⊂ U C ( U, U r K ) ⊗ Z D ( U, U r K ) . The local Poincar´e duality properties of topological manifolds do not directly suggestthe above definition of C ⊠ D , but rather an asymmetric definition, as follows. Definition 5.3.
For objects C and D of D , let C ⊠ ? D = holim U ⊂ X open K ⊂ U closed C ( U, U r K ) ⊗ Z D ( U ) . Remark . Both C ⊠ D and C ⊠ ? D are contractible if either C or D are in D ′′ . Lemma 5.5.
The specialization map C ⊠ D → C ⊠ ? D (obtained by specializing to K = X in the formula for C ⊠ D ) induces an isomorphism in homology.Proof. We begin with an informal argument. Let ξ be an n -cycle in C ⊠ D . For K , K ⊂ X closed, U ⊂ X open and K ∩ K contained in U , abbreviate F ( U, K , K ) = C ( U, U r K ) ⊗ Z D ( U, U r K ) . Choose K +1 ⊂ U closed, K − ⊂ X closed so that K = K +1 ∪ K − and K ∩ K − = ∅ . Thenwe have a commutative diagram F ( U, K +1 , K ) / / F ( U, K +1 ∩ K − , K ) F ( U, K − , K ) o o F ( U, K +1 , K ) / / = O O = (cid:15) (cid:15) F ( U, K +1 ∩ K − , K ) = O O = (cid:15) (cid:15) F ( U r K , K − , K ) = 0 o o O O (cid:15) (cid:15) F ( U, K +1 , K ) / / F ( U, K +1 ∩ K − , K ) F ( U r K , K +1 ∩ K − , K ) = 0 o o F ( U, K +1 , X ) / / O O F ( U, K +1 ∩ K − , X ) O O F ( U r K , K +1 ∩ K − , X ) o o O O By the sheaf properties, the coordinate of ξ in F ( U, K , K ) is sufficiently determined bythe projection of ξ to the homotopy inverse limit of the top row of the diagram. By diagramchasing, this is sufficiently determined by the projection of ξ to the homotopy inverse limitof the bottom row. But that information is stored in the image of ξ in C ⊠ ? D .The argument can be formalized as follows. For fixed open U , closed K and K with K ∩ K contained in U , we consider the poset P of “decompositions” K = K +1 ∪ K − where K +1 ⊂ U and K − are closed, K ∩ K − = ∅ . The ordering is such that ( J +1 , J − ) ≤ ( K +1 , K − ) if andonly if J +1 ⊂ K +1 and J − ⊂ K − . We need to know that B P is contractible. To see this let Q be the poset of all closed neighborhoods of K ∩ K in K ∩ U . Then ( K +1 , K − ) K +1 is a functor v : P → Q . Fixing some J ∈ Q , let P J be the poset of all ( K +1 , K − ) ∈ P with K +1 ⊂ J . This contains as a terminal sub-poset the set of all ( K +1 , K − ) with K +1 = J , andthe latter is clearly (anti-)directed. Hence B P J is contractible. This verifies the hypothesesin Quillen’s theorem A for the functor v , so that Bv : B P → B Q is a homotopy equivalence.But Q is again directed, so B Q is contractible. Therefore B P is contractible. Now we canwrite C ⊠ D = holim U,K ,K F ( U, K , K ) ≃ holim U,K ,K holim K +1 ,K − F ( U, K , K ) ∼ = holim U,K +1 ,K − ,K F ( U, K +1 ∪ K − , K ) ≃ holim U,K +1 ,K − ,K holim (cid:2) F ( U, K +1 , K ) → F ( U, K +1 ∩ K − , K ) ← F ( U, K − , K ) (cid:3) . (The usual conventions apply: K , K closed in X , with K ∩ K contained in the open set U ,and K = K − ∪ K +1 is a decomposition of the type we have just discussed.) Using that, we ONTRYAGIN CLASSES 13 obtain from the rectangular twelve term diagram above a map g : H ∗ ( C ⊠ ? D ) → H ∗ ( C ⊠ D ).Indeed an n -cycle in C ⊠ ? D determines an n -cycle inholim U,K +1 ,K − ,K holim (cid:2) F ( U, K +1 , X ) → F ( U, K +1 ∩ K − , X ) ← F ( U r K , K +1 ∩ K − , X ) (cid:3) and we use the vertical arrows in the twelve-term diagram to obtain an n -cycle inholim U,K +1 ,K − ,K holim (cid:2) F ( U, K +1 , K ) → F ( U, K +1 ∩ K − , K ) ← F ( U, K − , K ) (cid:3) whose homology class is well defined. By construction, g : H ∗ ( C ⊠ ? D ) → H ∗ ( C ⊠ D ) is leftinverse to the projection-induced map H ∗ ( C ⊠ D ) → H ∗ ( C ⊠ ? D ). But it is clearly alsoright inverse (specialize to K = X and then K − = ∅ in the diagrams above). (cid:3) Example 5.6.
Let X be a compact ENR. For open U in X , let C ( U ) be the singularchain complex of U , regarded as a subcomplex of C . By lemma 3.6, applied to the identity X → X , this functor U C ( U ) satisfies the sheaf type condition and the two finitenessconditions. We construct a “canonical” map ∇ : C ( X ) −→ C ⊠ C .
To start with we have the following diagram: C ( X ) −−−−→ holim U,K C ( X, X r K ) ≃ ←−−−− holim U,K C ( U, U r K )where U and K are open in X and closed in X , respectively, with K ⊂ U . The first arrowis induced by the quotient maps C ( X ) → C ( X, X r K ) and the second map is induced bythe inclusions C ( U, U r K ) → C ( X, X r K ) which are chain homotopy equivalences by thesheaf property. Inversion of the second arrow in the diagram gives us a map γ : C ( X ) −→ holim U,K C ( U, U r K )well defined up to contractible choice (in particular, well defined up to chain homotopy).Next we make use of a certain chain map ζ : holim U,K C ( U, U r K ) −→ C ⊠ C .
This is determined by the compositions C ( U, U r ( K ∩ K )) −−−−→ sing. chain cx. of ( U × U, U × U r ( K × U ∪ U × K )) y C ( U, U r K ) ⊗ C ( U, U r K )(for closed K , K ⊂ X with intersection K ∩ K ⊂ U ), where the first arrow is induced bythe diagonal map and the second arrow is an Eilenberg-Zilber map. Now we define ∇ = ζγ : C ( X ) −→ C ⊠ C .
This is a refinement of the standard Eilenberg-Zilber-Alexander-Whitney diagonal chainmap C ( X ) → C ( X ) ⊗ C ( X ), which we can recover by composing with the projection (aliasspecialization) from C ⊠ C to C ( X ) ⊗ C ( X ).If X is an oriented closed topological n -manifold and ω ∈ C ( X ) is an n -cycle representinga fundamental class for X , then ∇ ( ω ) is an n -cycle in C ⊠ C which, as we shall see inproposition 5.8 below, is “nondegenerate”. This reflects the fact that not only X , but alsoeach open subset of X satisfies a form of Poincar´e duality. Example 5.7.
Let (
X, Y ) be a pair of compact ENRs. For open U in X , let C ( U ) be thesingular chain complex of U and let D ( U ) be the singular chain complex of U ∩ Y . Bylemma 3.6, both C and D satisfy the sheaf type condition and the two finiteness conditions.A straightforward generalization of the previous example gives ∇ : C ( X ) /D ( X ) −→ ( C ⊠ C ) (cid:14) ( D ⊠ D ) . If X is an oriented compact topological n -manifold with boundary Y , and ω ∈ C ( X ) /D ( X )is an n -cycle representing a fundamental class for the pair ( X, Y ), then ∇ ( ω ) is an n -cyclein ( C ⊠ C ) (cid:14) ( D ⊠ D ) whose image in C ⊠ ( C/D ) is nondegenerate (proposition 5.8 below).
Proposition 5.8.
Let C and D be objects of C . Let [ ϕ ] ∈ H n ( C ⊠ D ) and suppose that, forevery open U ⊂ X and every j ∈ Z , the map colim cpct K ⊂ U H n − j C ( U, U r K ) −→ H j D ( U ) , slant product with the coordinate of ϕ in C ( U, U r K ) ⊗ D ( U ) , is an isomorphism. Then [ ϕ ] is nondegenerate in the following sense: for any E in D , the map f f ∗ [ ϕ ] is anisomorphism from [ D, E ] , the morphism set in C / C ′′ ∼ = HC ′ , to H n ( C ⊠ E ) .Proof. The idea is very simple. Given [ ψ ] ∈ H n ( C ⊠ E ), the slant product with [ ψ ] gives ushomomorphisms colim closed K ⊂ U H n − j C ( U, U r K ) −→ H j E ( U )for j ∈ Z . By our assumption on [ ϕ ], we have colim K H n − j C ( U, U r K ) ∼ = H j D ( U ) and sowe get homomorphisms H j D ( U ) −→ H j E ( U )for all j ∈ Z , naturally in U . It remains “only” to construct a morphism f ψ : D → E in D or in HD / HD ′′ ∼ = HD ′ inducing these homomorphisms of homology groups.By lemma 5.5 or otherwise, we may represent the class [ ψ ] by an n -cycle ψ ∈ C ⊠ ? E = holim U ⊂ X open K ⊂ U closed C ( U, U r K ) ⊗ Z E ( U ) . We shall show first of all that this n -cycle determines a chain map ψ ad : hocolim K ⊂ U C ( X, X r K ) n −∗ −→ holim V ⊃ U E ( V )natural in the variable U . In fact the source of this map is clearly a covariant functor ofthe variable U , by extension of the indexing poset . The target is homotopy equivalent to E ( U ), and we regard it as a covariant functor of U by restriction of the indexing poset . Moreprecisely, if U ⊂ U , then the poset of open subsets of X containing U is contained in thesubposet of open subsets of X containing U . Corresponding to that inclusion of posets wehave a projection map holim V ⊃ U E ( V ) −→ holim V ⊃ U E ( V )and that is what we use.To give a precise description of ψ ad now, we fix a string K ⊂ K ⊂ · · · ⊂ K r of compactsubsets of X , and a string of open subsets U s ⊂ U s − ⊂ · · · ⊂ U with K r ⊂ U s . Let ℓ ∆ r and ℓ ∆ s be the cellular chain complexes of ∆ r and ∆ s (the ℓ is for linearization ). We candefine ψ ad by associating to every choice of two such strings a chain map ℓ ∆ r ⊗ Z C ( X, X r K ) n −∗ −→ hom Z ( ℓ ∆ s , E ( U )) , ONTRYAGIN CLASSES 15 or equivalently, a chain map ℓ ∆ s ⊗ Z ℓ ∆ r −→ C ( X, X r K ) ⊗ Z E ( U )of degree n . (As the input strings vary, these chain maps are subject to obvious compati-bility conditions.) But in fact the coordinate of ψ ∈ C ⊠ ? E corresponding to the strings( K , . . . , K r ) and ( U s , . . . , U ) is a chain map ℓ ∆ s ⊗ ℓ ∆ r −→ C ( U , U r K ) ⊗ E ( U )of degree n . We need only compose with the inclusion-induced maps E ( U ) ⊗ D ( U, U r K r ) −→ E ( U ) ⊗ D ( X, X r K )to get the data we need.Now abbreviate P C ( n −∗ ) ( U ) = hocolim K ⊂ U C ( X, X r K ) n −∗ ,P E ( U ) = holim V ⊃ U E ( V ) ,P D ( U ) = holim V ⊃ U D ( V )(writing P for provisional ). Then we have a homotopy commutative diagram of naturaltransformations of functors on O ( X ), D ←−−−− C ( n −∗ ) −−−−→ E y y y P D ϕ ad ←−−−− P C ( n −∗ ) ψ ad −−−−→ P E as follows. The maps E → P E and D → P D are obvious (diagonal) constructions. Wenote that E ( U ) → P E ( U ) and D ( U ) → P D ( U ) are homology equivalences for all U . Theobject C ( n −∗ ) is chosen in C ′ and the map from it to P C ( n −∗ ) induces homology equivalences C ( n −∗ ) ( U ) → P C ( n −∗ ) ( U ) for all U by construction. (This uses lemma 4.6.) The arrows inthe top row are then constructed to make the diagram homotopy commutative. (This useslemma 4.7.) The horizontal arrows in the left-hand square of the diagram are homologyequivalences (for every choice of input U ) and consequently the arrow from C ( n −∗ ) to D is an isomorphism in C / C ′′ . We choose an inverse for it and compose with C ( n −∗ ) → E toobtain the desired element [ f ψ ] ∈ [ D, E ]. Finally, it is just a matter of inspection to see thatthe rule [ ψ ] [ f ψ ] is inverse to the map from [ D, E ] to H n ( C ⊠ E ) given by slant productwith [ ϕ ]. (cid:3) Example 5.9.
Recall that the natural Eilenberg-Zilber mapsg ch cx of ( Y × Z ) −→ (sg ch cx of Y ) ⊗ (sg ch cx of Z )(for arbitrary spaces Y and Z ) admits a refinement to a natural equivariant chain map W ⊗ (sg ch cx of ( Y × Z )) −→ (sg ch cx of Y ) ⊗ (sg ch cx of Z )where W is a free resolution of Z as a trivial module over the group ring Z [ Z / equivariance means that W ⊗ (sg ch cx of Y × Z ) −−−−→ (sg ch cx of Y ) ⊗ (sg ch cx of Z ) T ⊗ perm. y ∼ = perm. y ∼ = W ⊗ (sg ch cx of Z × Y ) −−−−→ (sg ch cx of Z ) ⊗ (sg ch cx of Y ) commutes for all Y and Z , where T is the generator of Z / W and “perm.” standsfor a permutation of the factors Y and Z .Applying this in the situation of example 5.6, we deduce immediately that ∇ : C ( X ) −→ C ⊠ C has a refinement to ∇ h Z / : C ( X ) −→ ( C ⊠ C ) h Z / . Suppose now that X is a closed topological manifold, and ω ∈ C ( X ) is a fundamentalcycle. Then with the nondegeneracy property which we have already established, we cansay informally that ( C, ∇ h Z / ( ω )) is an n -dimensional SAPC in C (slightly against the rules,since we have not established that every object in C has a dual).Similarly, in the situation and notation of example 5.7, we obtain ∇ h Z / ( ω ), an n -cyclein ( C ⊠ C ) h Z / / ( D ⊠ D ) h Z / . With the nondegeneracy property which we have alreadyestablished, this allows us to say informally that (( C, D ) , ∇ h Z / ( ω )) is an SAP pair.There are “economy” versions of C ⊠ D other than C ⊠ ? D . Suppose that Q is acollection of closed subsets of X with the following properties. • Q is closed under finite intersections (i.e., for Q and Q in Q , we have Q ∩ Q ∈ Q ) ; • for every compact subset K of X and open U ⊂ X containing K , there exist r ≥ Q , . . . , Q r ∈ Q such that K ⊂ r [ i =1 Q r ⊂ U .
For C and D in D we define C ⊠ ?? D = holim U ∈O ( X ) K ,K ∈Q K ∩ K ⊂ U C ( U, U r K ) ⊗ Z D ( U, U r K ) . This depends on Q , not just on C and D , but in practice it will be clear what Q is. Lemma 5.10.
The specialization map C ⊠ D −→ C ⊠ ?? D is a chain homotopy equivalence.Proof. Step 1: We assume that X is compact. Let Q ′ be the collection of all subsets of X which are finite unions of subsets K ∈ Q . The specialization map C ⊠ D −→ C ⊠ ?? D is a ONTRYAGIN CLASSES 17 composition of two specialization mapsholim U ∈O ( X ) K ,K closed K ∩ K ⊂ U C ( U, U r K ) ⊗ Z D ( U, U r K ) f (cid:15) (cid:15) holim U ∈O ( X ) K ,K ∈Q ′ K ∩ K ⊂ U C ( U, U r K ) ⊗ Z D ( U, U r K ) g (cid:15) (cid:15) holim U ∈O ( X ) K ,K ∈Q K ∩ K ⊂ U C ( U, U r K ) ⊗ Z D ( U, U r K ) . We are going to show that both of these are homotopy equivalences. For the specializationmap f , it suffices to observe that, by our assumptions on Q , the triples ( U, K , K ) with K , K ∈ Q ′ and K ∩ K ⊂ U form an initial sub-poset in the poset of all triples ( U, K , K )with K , K closed and K ∩ K ⊂ U . This refers to the usual ordering,( U, K , K ) ≤ ( V, L , L ) ⇔ U ⊂ V and K ⊃ L , K ⊃ L . For the specialization map g , it suffices by [DwK, 9.7] to show that for open U ⊂ X and L , L in Q ′ with L ∩ L ⊂ U , the canonical map C ( U, U r L ) ⊗ Z D ( U, U r L ) −→ holim K ,K ∈Q K ⊂ L , K ⊂ L C ( U, U r K ) ⊗ Z D ( U, U r K )is a chain homotopy equivalence. But this is true by lemma 3.9. ( Some details : The targetof this map can also be described, up to homotopy equivalence, as a double homotopy limitholim R holim K ,K ∈R K ⊂ L , K ⊂ L C ( U, U r K ) ⊗ D ( U, U r K )where R runs through the finite subsets of Q which are “large enough”. By large enough we mean that there exist K , K , . . . , K r and K , K , . . . , K s in R such that r [ i =1 K i = L , r [ j =1 K j = L . For fixed R , we have an Alexander-Whitney type homotopy equivalenceholim K ,K ∈R K ⊂ L , K ⊂ L C ( U, U r K ) ⊗ D ( U, U r K ) ≃ (cid:18) holim K ∈R K ⊂ L C ( U, U r K ) (cid:19) ⊗ (cid:18) holim K ∈R K ⊂ L D ( U, U r K ) (cid:19) natural in R . By lemma 3.9 and because R is large enough, the projections C ( U, U r L ) −→ holim K ∈R K ⊂ L C ( U, U r K ) ,D ( U, U r L ) −→ holim K ∈R K ⊂ L D ( U, U r K )are homotopy equivalences. Putting these facts together, we see thatholim K ,K ∈Q K ⊂ L , K ⊂ L C ( U, U r K ) ⊗ Z D ( U, U r K )is homotopy equivalent to the homotopy inverse limit of a constant functor, R 7→ C ( U, U r K ) ⊗ Z D ( U, U r K ) . Since the poset of all R is directed, the homotopy inverse limit is homotopy equivalent tothe unique value of that functor.)Step 2: X is arbitrary (but still locally compact Hausdorff and separable). Choose a compact Y ⊂ X which belongs to Q and is a neighborhood for the support of C and for the supportof D . Let Q Y = { Q ∈ Q | Q ⊂ Y } , a collection of compact subsets of Y which is closed under finite intersections. For open U ⊂ Y put C Y ( U ) = C ( U ∪ ( X r Y )) , D Y ( U ) = D ( U ∪ ( X r Y )) . Now we have a commutative diagram of specialization maps C ⊠ D / / (cid:15) (cid:15) C ⊠ ?? D (cid:15) (cid:15) C Y ⊠ D Y / / C Y ⊠ ?? D Y using Q Y to define the lower row. By step 1, the lower horizontal arrow is a homotopyequivalence. It is therefore enough to show that the two vertical arrows are homotopyequivalences. This follows easily from the fact that the inclusion of posets ι : W → O ( X )has a left adjoint, where O ( X ) consists of all open subsets of X and W consists of all opensubsets of X containing X r Y . The left adjoint is given by U λ ( U ) = U ∪ ( X r Y ).Note also that the inclusion-induced maps C ( U ) ∼ = C ( λ ( U )) and D ( U ) ∼ = D ( λ ( U )) areisomorphisms. Thus, λ induces maps C Y ⊠ D Y → C ⊠ D and C Y ⊠ ?? D Y → C ⊠ ?? D whichare homotopy inverses for the vertical arrows in our square. The homotopies are inducedby natural transformations, the “unit” and the “counit” of the adjunction of ι and λ . (cid:3) Products
Let X and Y be locally compact Hausdorff and separable spaces. Definition 6.1.
For C in C X and D in C Y , the tensor product C ⊗ D of C and D is theordinary tensor product of chain complexes C ⊗ Z D , with the system of subcomplexesdefined by ( C ⊗ D )( W ) = X U,VU × V ⊂ W C ( U ) ⊗ Z D ( V )for W ∈ O ( X × Y ). ONTRYAGIN CLASSES 19
We are aiming to show that C ⊗ D is in C X × Y . This is surprisingly hard. We beginwith two lemmas. Lemma 6.2.
For i ∈ { , , . . . , k } let U i be open in X and let V i be open in Y . For nonempty S ⊂ { , , . . . , k } put U S = T λ ∈ S U λ and V S = T λ ∈ S V λ . The following map (induced byobvious inclusions) is a homotopy equivalence: hocolim ∅6 = S ⊂{ , ,...,k } C ( U S ) ⊗ Z D ( V S ) −→ k X i =1 C ( U i ) × D ( V i ) , where the sum P ki =1 is taken inside C ( X ) ⊗ Z D ( Y ) .Proof. We proceed by induction on k . The square of inclusion maps P k − i =1 C ( U i ∩ U k ) ⊗ D ( V i ∩ V k ) −−−−→ C ( U k ) ⊗ D ( V k ) y yP k − i =1 C ( U i ) ⊗ D ( V i ) −−−−→ P ki =1 C ( U i ) ⊗ D ( V i ) ( ∗ )is a homotopy pushout square. Indeed, it is a pushout square in which the horizontalarrows (in fact, also the vertical arrows) are cofibrations, i.e., split injective as maps ofgraded abelian groups. The pushout property can be verified in terms of bases: each of thefour terms in the square is the graded free abelian group generated by a certain graded set.Next, for nonempty S ∈ { , . . . , k − } , write E ( S ) = C ( U S ) × D ( V S ). Then it is clear thathocolim S E ( S ∪ k ) −−−−→ hocolim S E ( k ) y y hocolim S E ( S ) −−−−→ hocolim S hocolim( E ( S ) ← E ( S ∪ k ) → E ( k )) , ( ∗∗ )where S refers to nonempty subsets of { , . . . , k − } , commutes up to a preferred homotopy h and, as such, is a homotopy pushout square. The square ( ∗∗ ) maps to ( ∗ ) by a forgetfulmap which also takes the homotopy h to zero. By inductive hypothesis, three of the fourarrows which constitute this map ( ∗∗ ) → ( ∗ ) between squares are homotopy equivalences,and therefore all are homotopy equivalences. Now it only remains to show that the canonicalmap hocolim ∅6 = S ⊂{ , ,...,k − } hocolim( E ( S ) ← E ( S ∪ k ) → E ( k )) y hocolim ∅6 = S ⊂{ , ,...,k } E ( S )is a homotopy equivalence. We have dealt with this kind of task before, in the proof oflemma 3.9 and lemma 3.10, and it can be dealt with in the same way here. (cid:3) Lemma 6.3.
In the situation of lemma 6.2, let W be the union of the sets U i × V i for i = 1 , . . . , k . Then the inclusion k X i =1 C ( U i ) ⊗ Z D ( V i ) −→ ( C ⊗ D )( W ) is a homotopy equivalence. Proof.
Step 1: We assume to begin with that W itself has the form U × V for some U openin X and some V open in Y . It is easy to construct finite open coverings { U ′ λ | λ ∈ Λ } of U , and { V ′ γ | γ ∈ Γ } of V , such that every open set U i × V i is a union of (some) of the sets U ′ λ × V ′ µ . Now the composition of inclusions X ( λ,µ ) C ( U ′ λ ) ⊗ D ( V ′ µ ) yX i C ( U i ) ⊗ D ( V i ) y C ( U × V )is a homotopy equivalence, because the source can be written as X λ C ( U ′ λ ) ! ⊗ Z X µ C ( V ′ µ ) ! and we are assuming the sheaf type condition for C and D . Therefore it remains only toshow that the first of these inclusions admits a homotopy left inverse. Using lemma 6.2, wemay replace P i C ( U i ) ⊗ D ( V i ) byhocolim ∅6 = T ⊂{ ,...,k } C ( U T ) ⊗ D ( V T ) . In that expression, C ( U T ) can be replaced by the sum of the C ( U ′ R ) for U ′ R ⊂ U T , where R is a nonempty subset of Λ. Similarly D ( V T ) can be replaced by the sum of the D ( V ′ S ) for V ′ S ⊂ V T , where S is a nonempty subset of Γ. (Here we use the sheaf type conditions for C and D again.) After these modifications, we have an obvious projection map from thathomotopy colimit to X ( λ,µ ) C ( U ′ λ ) ⊗ D ( V ′ µ ) . This provides the required left homotopy inverse.Now we look at the general case. Let W = S ki =1 U i × V i as given. Let U k +1 ⊂ X and V k +1 ⊂ Y be open and suppose U k +1 ⊂ V k +1 ⊂ W . It is enough to show that the inclusion k X i =1 C ( U i ) ⊗ D ( V i ) −→ k +1 X i =1 C ( U i ) ⊗ D ( V i )is a homotopy equivalence. (For then we can repeat the process by adding on as many terms C ( U ′ ) ⊗ D ( V ′ ) with U ′ × V ′ ⊂ W as we like, and thereby approximate ( C ⊗ D )( W ).) Thereis a pushout square P ki =1 C ( U i ∩ U k +1 ) ⊗ D ( V i ∩ V k +1 ) −−−−→ C ( U k +1 ) ⊗ D ( V k +1 ) y yP ki =1 C ( U i ) ⊗ D ( V i ) −−−−→ P k +1 i =1 C ( U i ) ⊗ D ( V i )in which the horizontal arrows are cofibrations. It follows that the square is a homotopypushout square. By step 1, the upper horizontal arrow is a homotopy equivalence. Thereforethe lower horizontal arrow is a homotopy equivalence. (cid:3) ONTRYAGIN CLASSES 21
Lemma 6.4. If C belongs to C X and D belongs to C Y , then C ⊗ D belongs to C X × Y .Proof. For the sheaf condition, suppose that W ⊂ X × Y is open and W = S W α . Then W is the union of all open sets U i × V i which are contained in some W α , and therefore theinclusion X α ( C ⊗ D )( W α ) = X i C ( U i ) ⊗ D ( U i ) −→ ( C ⊗ D )( W )is a homotopy equivalence by lemma 6.3 and passage to the direct limit.For finiteness condition (ii), suppose that C has support in a compact subset K of X and D has support in a compact subset L of Y . Then it is clear that C ⊗ D has support in K × L ⊂ X × Y .This leaves finiteness condition (i) to be established. We recall what it requires. We have tofind an integer c ≥ W ⊂ W ′ in X × Y , where W has compact closurein W ′ , the inclusion ( C ⊗ D )( W ) → ( C ⊗ D )( W ′ ) factors through a chain complex of f.g.free abelian groups whose i -th chain group is zero whenever | i | > c . Let a, b ∈ Z be thecorresponding integers for C and D . Let us provisionally say that an open set W in X × Y is good if W has compact closure in X and, for every open W ′′ in X containing the closure¯ W of W , there exists another open W ′ with compact closure in X such that W ⊂ ¯ W ⊂ W ′ ⊂ ¯ W ′ ⊂ W and the inclusion map ( C ⊗ D )( W ′ ) → ( C ⊗ D )( W ′′ ) factors through a boundedchain complex of f.g. free abelian groups.Then it is easy to verify: • any W of the form W = U × V , with compact closure in X × Y , is good; • if W , W and W ∩ W are good open subsets of X × Y , then W ∪ W is good.It follows that if W is any finite union of subsets of the form U × V , where U and V areopen in X and Y , respectively, with compact closures, then W is good. While that doesnot prove all we need, it will now be sufficient to show that, for any W open in X × Y ,the chain complex ( C ⊗ D )( W ) is homotopy equivalent to a chain complex of free abeliangroups concentrated in degrees between − ab and ( a +2)( b +2)+1. To that end, we note thatfor any open U in X , the chain complex C ( U ) is homotopy equivalent to a chain complexof free abelian groups concentrated in degrees between − a and a + 1, as it is homotopyequivalent to a sequential homotopy colimit of chain complexes concentrated in degreesbetween − a and a . Consequently, for open U ′ , U ′′ in X with U ′ ⊂ U ′′ , the chain complex C ( U ′ , U ′′ ) = C ( U ′ ) /C ( U ′′ ) is homotopy equivalent to a chain complex of free abelian groupsconcentrated in degrees between − a and a + 2. Similar remarks apply to D in place if C .If W is open in X × Y and is a finite union of subsets of the form U α × V α , then we mayreplace ( C ⊗ D )( W ) by the homotopy equivalent subcomplex X α C ( U α ) ⊗ D ( V α ) . This admits a finite filtration by subcomplexes such that the subquotients of the filtrationhave the form C ( U ′ , U ′′ ) ⊗ Z D ( V ′ , V ′′ )for some open U ′ , U ′′ in X and V ′ , V ′′ in Y , with U ′ ⊂ U ′′ and V ′ ⊂ V ′′ . It is thereforehomotopy equivalent to a chain complex of free abelian groups concentrated in degreesbetween − ab and ( a + 2)( b + 2). Finally, an arbitrary open W in X × Y can be written as amonotone union of subsets W i where each W i is a finite union of open subsets of the form U α × V α . Since each ( C ⊗ D )( W i ) is homotopy equivalent to a chain complex of free abeliangroups concentrated in degrees between − ab and ( a + 2)( b + 2), it follows that ( C ⊗ D )( W )is homotopy equivalent to a chain complex of free abelian groups concentrated in degreesbetween − ab and ( a + 2)( b + 2) + 1. (cid:3) Corollary 6.5. If C belongs to D X and D belongs to D Y , then C ⊗ D belongs to D X × Y .Proof. It is clear from the definition that, for fixed C , the functor D C ⊗ D from C X to C X × Y respects weak equivalences and short exact sequences. The same is true if we fix D and allow C to vary. Therefore it is enough to prove the claim in the special case where C and D are obtained from maps f : ∆ k → X and g : ∆ ℓ → Y , so that C ( U ) and D ( V ) are thesingular chain complexes of f − ( U ) and g − ( V ), respectively, for U open in X and V openin Y . Let E in C X × Y be the object obtained from f × g : ∆ k × ∆ ℓ → X × Y , so that E ( W )is the singular chain complex of ( f × g ) − ( W ), for W open in X × W . It is easy to showthat E belongs to D X × Y . There is an easy Eilenberg-Zilber type map C ⊗ D −→ E in C X × Y . For open W in X × Y of the form W = U × V with U open in X and V open in Y , this specializes to a map of chain complexes( C ⊗ D )( W ) −→ E ( W )which is a chain homotopy equivalence by the very Eilenberg-Zilber theorem. It follows thenfrom the sheaf properties that ( C ⊗ D ) → E ( W ) is always a chain homotopy equivalence,for arbitrary open W in X × Y . Consequently C ⊗ D belongs to D X × Y . (cid:3) Proposition 6.6.
Let C and E be objects of C X . Let D and F be objects of C Y . Thenthe following specialization map is a homotopy equivalence (and a “fibration”, i.e., splitsurjective as a map of graded abelian groups): ( C ⊗ D ) ⊠ ( E ⊗ F ) = holim W,P ,P ( C ⊗ D )( W, W r P ) ⊗ ( E ⊗ F )( W, W r P ) (cid:15) (cid:15) holim W = U × VP = J × K P = J × K ( C ⊗ D )( W, W r P ) ⊗ ( E ⊗ F )( W, W r P ) . Proof.
As in the proof of lemma 5.10, there is no loss of generality in assuming that X and Y are both compact. In that case we may replace the target of our specialization map byholim WP = J × K P = J × K ( C ⊗ D )( W, W r P ) ⊗ ( E ⊗ F )( W, W r P ) , dropping the condition W = U × V . (This makes no difference to the homotopy typebecause, in the poset of triples ( W, P , P ) where P = J × K and P = J × K , thosetriples which have W = U × V for some U and V form an “initial” sub-poset.) Now ourmap has the form ( C ⊗ D ) ⊠ ( E ⊗ F ) −→ ( C ⊗ D ) ⊠ ?? ( E ⊗ F )as in lemma 5.10, and it is a homotopy equivalence by that same lemma. (cid:3) ONTRYAGIN CLASSES 23
Definition 6.7.
We construct a map( C ⊠ D ) ⊗ ( E ⊠ F ) −→ ( C ⊗ E ) ⊠ ( D ⊗ F )by composing the following: ( C ⊠ D ) ⊗ ( E ⊠ F ) (cid:15) (cid:15) holim W = U × VP = J × K P = J × K ( C ⊗ D )( W, W r P ) ⊗ ( E ⊗ F )( W, W r P ) (cid:15) (cid:15) ( C ⊗ E ) ⊠ ( D ⊗ F )The second arrow is a right inverse for the chain map in proposition 6.6. The first arrow isinduced by isomorphisms (cid:0) C ( U, U r J ) ⊗ D ( U, U r J ) (cid:1) ⊗ (cid:0) E ( V, V r K ) ⊗ F ( V, V r K ) (cid:1) ∼ = (cid:15) (cid:15) (cid:0) ( C ⊗ E )( W, W r P ) (cid:1) ⊗ (cid:0) ( D ⊗ F )( W, W r P ) (cid:1) where W = U × V and P i = J i × K i for i = 1 ,
2. We describe the composite map informallyas ϕ ⊗ ψ ϕ ¯ ⊗ ψ , where ϕ and ψ are chains in C ⊠ D and E ⊠ F , respectively. Definition 6.8.
Assume C = D and E = F in definition 6.7. We construct a map( C ⊠ C ) h Z / ⊗ ( E ⊠ E ) h Z / −→ (( C ⊗ E ) ⊠ ( C ⊗ E )) h Z / by composing the following: ( C ⊠ C ) h Z / ⊗ ( E ⊠ E ) h Z / (cid:15) (cid:15) (cid:18) holim W = U × VP = J × K P = J × K ( C ⊗ E )( W, W r P ) ⊗ ( C ⊗ E )( W, W r P ) (cid:19) h Z / (cid:15) (cid:15) (cid:0) ( C ⊗ E ) ⊠ ( C ⊗ E ) (cid:1) h Z /
24 ANDREW RANICKI AND MICHAEL WEISS (details as in definition 6.7, with superscripts h Z / ϕ ⊗ ψ ϕ ¯ ⊗ ψ , where ϕ and ψ are chains in( C ⊠ C ) h Z / and ( E ⊠ E ) h Z / , respectively. Example 6.9.
In particular, suppose that (
C, ϕ ) and (
E, ψ ) are “symmetric objects” (notnecessarily Poincar´e ) of dimensions m and n respectively, in C X and C Y respectively, sothat ϕ is an m -cycle in ( C ⊠ C ) h Z / and ψ is an n -cycle in ( D ⊠ D ) h Z / . Then we have( C ⊗ D, ϕ ¯ ⊗ ψ ) , a symmetric object of dimension m + n in C X × Y . Example 6.10.
Let X and Y be compact ENRs. Let C be the functor taking an open U ⊂ X to the singular chain complex of U . Let E be the functor taking an open V ⊂ Y tothe singular chain complex of V . Let F be the functor taking an open W ⊂ X × Y to thesingular chain complex of W . By all the above, we have the following diagram C ( X ) ⊗ E ( Y ) ≃ (cid:15) (cid:15) ∇⊗∇ / / ( C ⊠ C ) h Z / ⊗ ( E ⊠ E ) h Z / (cid:15) (cid:15) F ( X × Y ) ∇ / / ( F ⊠ F ) h Z / . which is commutative up to a chain homotopy. (We could be more precise about that byspecifying a “contractible choice” of such chain homotopies.) The dotted arrow is given asin definition 6.8 by ϕ ⊗ ψ ϕ ¯ ⊗ ψ . We leave it to the reader to establish the homotopycommutativity. 7. Duality and decomposability
We turn to a discussion of duality, first in C , then in D and then in r D . Let C and D be objects of C which admit duals C ( −∗ ) and D ( −∗ ) . (In other words, the functors E H ( C ⊠ E ) and E H ( D ⊠ E ) on HD / HD ′′ are co-representable.) Fix nondegenerate0-cycles ϕ ∈ C ⊠ C ( −∗ ) , ψ ∈ D ⊠ D ( −∗ ) . Let f : C → D be any morphism. We can assume that D ( −∗ ) and C ( −∗ ) are in C ′′ . Choosea morphism g : D ( −∗ ) → C ( −∗ ) such that g ∗ ( ψ ) ∈ D ⊠ C ( −∗ ) is homologous to f ∗ ( ϕ ). Thenchoose a 1-chain ζ ∈ D ⊠ C ( −∗ ) such that dζ = g ∗ ( ψ ) − f ∗ ( ϕ ). Now for every E in C thesquare hom( C ( −∗ ) , E ) slant with ϕ (cid:15) (cid:15) g ∗ / / hom( D ( −∗ ) , E ) slant with ψ (cid:15) (cid:15) C ⊠ E f ∗ / / D ⊠ E is homotopy commutative (slant with ζ provides a homotopy), and the vertical arrows arehomology equivalences. Writing M ( g ) and M ( f ) for the mapping cone of g and f respectively,we have homotopy cofiber sequenceshom( M ( g ) , E ) / / hom( C ( −∗ ) , E ) g ∗ / / hom( D ( −∗ ) , E ) ,C ⊠ E f ∗ / / D ⊠ E / / M ( f ) ⊠ E .
Our homotopy commutative square therefore implies that the functor E H ( M ( f ) ⊠ E ) isagain co-representable, with representing object Σ − M ( g ). In particular the identity class in ONTRYAGIN CLASSES 25 H hom( M ( g ) , M ( g )) corresponds to some nondegenerate class [ λ ] ∈ H ( M ( f ) ⊠ M ( g )). Thisconstruction of [ λ ] shows also that, for every open U ∈ O ( X ) and closed K ⊂ X containedin U , we have a commutative diagram... ... H j +1 M ( f )( U, U r K ) O O slant with λ / / H j M ( g )( U ) O O H j C ( U, U r K ) O O slant with ϕ / / H j C ( −∗ ) ( U ) O O H j D ( U, U r K ) f ∗ O O slant with ψ / / H j D ( −∗ ) ( U ) g ∗ O O H j M ( f )( U, U r K ) O O slant with λ / / H j +1 M ( g )( U ) O O ... O O ... O O with exact columns. This leads us to the following conclusion. Lemma 7.1.
Objects of D have duals which are again in D . For E and F in D , an element [ λ ] ∈ H ( E ⊠ F ) is nondegenerate if and only if, for all open U in X , the slant product with [ λ ] is an isomorphism colim K H j E ( U, U r K ) −→ F ( U ) (where K runs through the closed subsets of X which are contained in U ).Proof. By the preceding discussion, it is enough to show that an object E of D constructed asin example 3.4 from a map f : ∆ k → X has a dual F in C , with nondegenerate [ λ ] ∈ H ( E ⊠ F )say, that F is again decomposable, and that the slant product with λ is an isomorphismcolim K H j E ( U, U r K ) −→ F ( U )for all U ∈ O ( X ). By example 5.7 and proposition 5.8, all that is true. (cid:3) Corollary 7.2.
Objects of r D have duals which are again in r D . For E and F in r D , anelement [ λ ] ∈ H ( E ⊠ F ) is nondegenerate if and only if, for all open U in X , the slantproduct with [ λ ] is an isomorphism colim K H j E ( U, U r K ) −→ F ( U ) . Proof.
Let E be an object in r D . We can assume that E is in r D ′ and that E admits a“complement” E ′ , also in r D ′ , so that E ′′ = E ⊕ E ′ belongs to D . Now E ′′ admits a dual,say F ′′ in D ′ , coupled to E ′′ by means of[ λ ′′ ] ∈ H ( E ′′ ⊠ F ′′ ) . The retraction map q : E ′′ → E ′′ (via E ) has a dual p : F ′′ → F ′′ , so that ( q ⊗ id) ∗ [ λ ′′ ] =(id ⊗ p ) ∗ [ λ ′′ ] ∈ H ( E ′′ ⊗ F ′′ ). As q is idempotent, p is idempotent up to homotopy. We can now produce a splitting F ′′ ≃ F ⊕ F ′ . Namely, for open U in X we let F ( U ) be thehomotopy colimit of F ′′ ( U ) p / / F ′′ ( U ) p / / F ′′ ( U ) p / / · · · and we let F ′ ( U ) be the homotopy colimit of F ′′ ( U ) id − p / / F ′′ ( U ) id − p / / F ′′ ( U ) id − p / / · · · . Then it is clear that F and F ′ belong to r D ′ and H ( E ′′ ⊠ F ′′ ) ∼ = H ( E ⊠ F ) ⊕ H ( E ′ ⊠ F ′ ) ⊕ H ( E ′ ⊠ F ) ⊕ H ( E ⊠ F ′ ) . The equation ( q ⊗ id) ∗ [ λ ′′ ] = (id ⊗ p ) ∗ [ λ ′′ ] shows that [ λ ′′ ] lives in H ( E ⊠ F ) ⊕ H ( E ′ ⊠ F ′ ) . Write [ λ ′′ ] = [ λ ] ⊕ [ λ ′ ] with [ λ ] ∈ H ( E ⊠ F ) and [ λ ′ ] ∈ H ( E ′ ⊠ F ′ ). It is straightforwardto show that [ λ ] ∈ H ( E ⊠ F ) is nondegenerate and that the slant product with [ λ ] is anisomorphism colim K H j E ( U, U r K ) −→ F ( U ) , because [ λ ′′ ] has the analogous properties. The universal property of [ λ ] now implies that,for arbitrary F ♯ in r D with [ λ ♯ ] ∈ H ( E ⊠ F ♯ ), the element [ λ ♯ ] is nondegenerate if and onlyif the slant product with it is an isomorphismcolim K H j E ( U, U r K ) −→ F ♯ ( U ) . (cid:3) Lemma 7.3.
The rule X
7→ D X is a covariant functor, preserving duality.Proof. Let f : X → Y be a map (between locally compact separable Hausdorff spaces). It isclear that, for C in D X , we have f ∗ D in D Y . For C and D in D X , there is a specializationmap C ⊠ D −→ f ∗ C ⊠ f ∗ D. We need to show that this takes nondegenerate classes in H ( C ⊠ D ) to nondegenerateclasses in H ( f ∗ C ⊠ f ∗ D ). In the case where X and Y are both compact, this follows fromthe nondegeneracy criterion given in lemma 7.1. (For U open in Y , every compact subsetof f − ( U ) is contained in some f − ( K ) for compact K ⊂ U .) Finally, because of finitenesscondition (ii) in definition 3.5, it is easy to reduce to a situation where X and Y are bothcompact. (cid:3) Corollary 7.4.
The rule X
7→ D X is a covariant functor, preserving duality. (cid:3) Proposition 7.5.
The tensor product D X × D Y −→ D X × Y is compatible with duality.Proof. Let
C, E be objects of D X and let D, F be objects of D Y . Let [ λ ] ∈ H ( C ⊠ E ) and[ µ ] ∈ H ( D ⊠ F ). Then we have [ λ ¯ ⊗ µ ] ∈ H (( C ⊗ D ) ⊠ ( E ⊗ F )). What we have to showis that if [ λ ] and [ µ ] are nondegenerate, then [ λ ¯ ⊗ µ ] is nondegenerate. This is a statementabout the three triangulated categories HD ′ X , HD ′ Y and HD ′ X × Y . For each of the threetriangulated categories we know that duality preserves exact triangles. We also know thatthe ⊠ product preserves exact triangles when one input variable is fixed. Hence, usingrepeated five lemma arguments, we can easily reduce the claim about the nondegeneracy of[ λ ¯ ⊗ µ ] to the special case where C and D are among the standard generators of D X and D Y ,respectively. That is, C is weakly equivalent to the object of D X obtained from some map f : ∆ k → X by the method of example 3.4, and D is weakly equivalent to the object of D X ONTRYAGIN CLASSES 27 obtained from some map g : ∆ ℓ → Y by the same method. (It seems better to say “weaklyequivalent” rather than “equal” because we might want to apply the resolution procedureof lemma 4.6 to obtain objects in D ′ X and D ′ Y , respectively.) In that case, we also have aclear idea what E and F are, and what [ λ ] and [ µ ] are. Namely, E is (up to desuspensions)the quotient of C by its “boundary” (the object obtained from f | ∂ ∆ k by the method ofexample 3.4), and F is (up to desuspensions) the quotient of D by its “boundary” (theobject obtained from g | ∂ ∆ ℓ by the method of example 3.4). Also, [ λ ] can be described asthe class of ∇ ( ω k ) where ω k is a relative fundamental cycle for the manifold-with-boundary∆ k , and [ µ ] can be described as the class of ∇ ( ω ℓ ) where ω ℓ is a relative fundamental cyclefor the manifold-with-boundary ∆ k .Next, C ⊗ D can be identified with the object obtained from f × g : ∆ k × ∆ ℓ −→ X × Y by the method of example 3.4. Also E ⊗ F can be identified (up to desuspensions) with thequotient of C ⊗ D by its “boundary”, which is the object obtained from f × g restrictedto ∂ (∆ k × ∆ ℓ ) by the method of example 3.4. Now example 6.10 implies that [ λ ¯ ⊗ µ ] canbe described as the class of ∇ ( ω k × ω ℓ ). Since ω k × ω ℓ is a relative fundamental cycle for∆ k × ∆ ℓ , this implies (with examples 5.6, 5.7 and proposition 5.8) that [ λ ¯ ⊗ µ ] is indeednondegenerate. (cid:3) The excisive signature
Lemma 8.1.
The functor X L • ( D X ) is homotopy invariant.Proof. It is enough to show that the maps X → X × [0 ,
1] given by x ( x,
0) and x ( x, π ∗ L • ( D X ) −→ L • ( D X × [0 , ) . That is easily done by using the 1-dimensional manifold with boundary [0 ,
1] and the corre-sponding SAPC in C [0 , , and tensor product with that, to produce appropriate bordisms. (cid:3) Theorem 8.2.
The functor X L • ( D X ) is excisive. In detail: (i) For open
U, V subset X with U ∪ V = X , the commutative square of inclusion-induced maps L • ( D U ∩ V ) −−−−→ L • ( D U ) y y L • ( D V ) −−−−→ L • ( D X ) . is homotopy (co)cartesian ; (ii) For a finite or countably infinite disjoint union X = ` X α , the inclusions X α → X induce a (weak) homotopy equivalence _ α L • ( D X α ) −→ L • ( D X ) . Proof.
Excision property (ii) is a straightforward consequence of finiteness property (ii).With corollary 4.15, the proof of excision property (i) can be given using a mechanismwhich is very nicely abstracted in a paper by Vogel [Vog, 1.18, 6.1]. (cid:3)
Theorem 8.3.
The relative homotopy groups of the inclusion L • ( D X ) → L • ( r D X ) arevector spaces over Z / . Proof.
Let K ( D ) be the Grothendieck group of D (with one generator [ C ] for each object C , a relation [ C ] ∼ [ D ] if C and D are weakly equivalent, and a relation [ C ] − [ D ] + [ E ] = 0for every short exact sequence C → D → E in D ). Define the Grothendieck group of r D similarly. Let ˜ K be the cokernel of the inclusion-induced map K ( D ) → K ( r D ). The group Z / L • ( D X ) → L • ( r D X ) can be described as a “Rothenberg” sequence: · · · → L n ( D X ) → L n ( r D X ) → b H n ( Z /
2; ˜ K ) → L n − ( D X ) → L n − ( r D X ) → · · · where b H ∗ denotes Tate cohomology. (cid:3) Remark . Lemma 8.1 and theorems 8.2 and 8.3 have analogues for quadratic L -theorywhich can be proved in the same way.If our locally compact Hausdorff separable space X is an ENR, then L • ( D X ) ≃ X + ∧ L • ( D pt ) = X + ∧ L • ( Z ) . This follows from homotopy invariance and the two excision properties by the standardarguments going back to Eilenberg and Steenrod. Here L • ( Z ) is the symmetric L -theoryspectrum of the ring Z (with the trivial involution). Remark . If X is the polyhedron of a simplicial complex L • ( D X ) has the homotopy typeof the spectrum L • ( Z , X ) constructed in [Ra2, §
10] from the ( Z , X )-category of [RaWe1]endowed with a chain duality. See also [Woolf], [RaWe2], [LM].If X is a compact oriented topological n -manifold with boundary ∂X , then the identitymap X → X determines by example 3.4 and example 5.7 an n -dimensional SAP pair in r D X (with boundary in r D ∂X ) which in turn determines an element in π n ( L • ( r D X ) , L • ( r D ∂X )) ∼ = Z [1 / π n ( L • ( D X ) , L • ( D ∂X )) ∼ = H n ( X, ∂X ; L • ( Z )) . Definition 8.6.
This element in π n ( L • ( r D X ) , L • ( r D ∂X )) is the excisive signature of ( X, ∂X ). Remark . If X is triangulable, then we can regard the excisive signature of ( X, ∂X ) asan element of π n ( L • ( D X ) , L • ( D ∂X )). In fact the excisive signature of a compact topologicalmanifold X with boundary, not necessarily triangulable, can always be regarded as anelement of π n ( L • ( D X ) , L • ( D ∂X )) . This follows easily from the fact that X × I n for sufficiently large n admits a handle decom-position. Unfortunately the proof of that fact (existence of handle decomposition) given e.g.in [KiSi] is hard and uses ideas which are quite closely related to Novikov’s original proofof the topological invariance of Pontryagin classes. For this reason we do not wish to usethe “handle decomposition” argument. We have already avoided it by introducing r D X andproving theorem 8.3. Remark . Let f : Y → X be a degree 1 normal map of closed n -dimensional topologicalmanifolds. By example 3.4 and example 5.7, the map f : Y → X and the identity mapid : X → X determine two n -dimensional SAP objects ( C ( f ) , ϕ ) and ( C (id) , ψ ) in r D X (and even in D X , by the previous remark). The map f induces a chain map C ( f ) → C (id)which respects the symmetric structures, so that there is a splitting up to weak equivalencein r D X or D X , ( C ( f ) , ϕ ) ≃ ( C (id) , ψ ) ⊕ ( K, ζ ) . ONTRYAGIN CLASSES 29
We expect that the nondegenerate symmetric structure ζ on K has a canonical refinementto a (nondegenerate) quadratic structure, determined by the bundle data which come withthe normal map f .9. The Poincar´e dual of the excisive signature
There is a rational homotopy equivalence L • ( Z ) ≃ Q _ i ≥ S i ∧ H Q , unique up to homotopy. For a compact oriented topological n -manifold X with boundary,the Poincar´e dual of the “rationalized” excisive signature of ( X, ∂X ) is therefore a class in M i ≥ H i ( X ; Q ) . We shall show that it is a characteristic class associated with the topological tangent bundleof X , a bundle with structure group TOP( n ). Lemma 9.1.
The suspension isomorphism H (pt; L • ( Z )) −→ H ( I, ∂I ; L • ( Z )) takes the unit to the excisive signature of ( I, ∂I ) . (cid:3) Proposition 9.2.
Let X be a compact oriented topological n -manifold X with boundaryand let Y = X × [0 , , so that Y /∂Y ∼ = Σ( X/∂X ) . The suspension isomorphism H n ( X, ∂X ; L • ( Z )) ⊗ Z [1 / −→ H n +1 ( Y, ∂Y ; L • ( Z )) ⊗ Z [1 / takes the excisive signature of ( X, ∂X ) to the excisive signature of ( Y, ∂Y ) .Proof. This follows from the previous lemma and the product formula in example 6.10. (cid:3)
Proposition 9.3.
Let X be a compact oriented topological n -manifold X with boundary, Y ⊂ X a compact codimension zero submanifold with locally flat boundary, Y ∩ ∂X = ∅ .Then, under the homomorphism H n ( X, ∂X ; L • ( Z )) ⊗ Z [1 / −→ H n ( Y, ∂Y ; L • ( Z )) ⊗ Z [1 / induced by the quotient map X/∂X → Y /∂Y , the excisive signature of ( X, ∂X ) maps to theexcisive signature of ( Y, ∂Y ) .Proof. This is a consequence of the naturality of ∇ in example 5.6. (cid:3) Proposition 9.4.
The Poincar´e dual of the (rationalized) excisive signature of a compactoriented manifold with boundary is a characteristic class Λ for euclidean bundles, evaluatedon the tangent (micro)bundle of the manifold. (The characteristic class Λ is defined foreuclidean bundles on compact ENRs, and is invariant under stabilisation, i.e., replacing aeuclidean bundle E → Y by E × R → Y .)Remark . While the construction of Λ as such is elementary, we use a technical fact fromgeometric topology to show that Λ(
T Y ) is Poincar´e dual to σ ( Y, ∂Y ) in the case where Y isa compact n -manifold. This fact is the existence of stable normal bundles for embeddingsof topological manifolds [Hi2]. Proof of proposition 9.4.
Let Y be a finite simplicial complex and let E → Y be a bundle on Y with fibers homeomorphic to R k . We would like to find a compact topological n -manifold X for some n , and a homotopy equivalence f : Y → X such that f ∗ T X is isomorphic to E × R n − k → Y , a stabilized version of E → Y . Assuming that a sufficiently canonical choiceof such an X and f can be made, we may then define the characteristic class associatedwith E on Y to be f ∗ of the Poincar´e dual of the excisive signature of ( X, ∂X ).For the first step of this program, we choose an embedding Y → R ℓ which is linear on eachsimplex of Y . Let Y r be a regular neighborhood of Y in R ℓ . Choose an extension of E toa euclidean bundle E r → Y r . Let X → Y r be the bundle of ( k + 1)-disks on Y r obtainedfrom E r → Y r by fiberwise one-point-compactification, followed by fiberwise join with apoint. Then X is a compact oriented manifold of dimension ℓ + k + 1. Let f : Y → X be thecomposition of the inclusion Y → Y r with any section of X → Y r . It is clear that f ∗ T X isidentified with E × R ℓ +1 → Y .We now define, as promised,Λ( E → Y ) = f ∗ ( u X ) ∈ M i ≥ H i ( Y ; Q )where u X ∈ L i ≥ H i ( X ; Q ) is the Poincar´e dual of the rationalized excisive signature σ ( X, ∂X ), in other words u X ∩ [ X, ∂X ] = σ ( X, ∂X ) . From proposition 9.2 we deduce that this is well defined, i.e., independent of the choice ofan ℓ and an embedding Y → R ℓ . (More precisely proposition 9.2 gives us the permissionto make ℓ as large as we like, and for large ℓ any two embeddings Y → R ℓ are isotopic.)From proposition 9.3, we deduce that Λ is a characteristic class. Namely, suppose thatwe have euclidean bundles E → Y and E ′ → Y ′ and a simplicial map g : Y → Y ′ suchthat g ∗ E ′ ∼ = E . Taking ℓ large, we can choose embeddings Y → R ℓ and Y ′ → R ℓ , withregular neighborhoods Y r and Y ′ r , in such a way that there is a codimension zero embedding g r : Y r → Y ′ r making the following diagram homotopy commutative: Y inclusion −−−−−→ Y r y g y g r Y ′ inclusion −−−−−→ Y ′ r . Now proposition 9.3 can be applied to the embedding g r and gives the desired conclusion,that g ∗ Λ( E ′ ) = Λ( E ) in L i ≥ H i ( Y ; Q ).Finally we can mechanically extend the definition of Λ to obtain a characteristic class definedfor euclidean bundles on compact ENRs. Indeed let Y be a compact ENR; then Y is a retractof some finite simplicial complex Y . Hence any euclidean bundle on Y extends to one on Y . We can evaluate the characteristic class Λ there, and pull back to the cohomology of Y . To show that this is well defined, use the following: if Y is a retract of a finite simplicialcomplex Y , and also a retract of a finite simplicial complex Y , then the union of Y and Y along Y is again an ENR. See [Hu].This is not the end of the proof, because we still have to show thatΛ( T Y ) ∩ ( Y, ∂Y ) = σ ( Y, ∂Y ) ∈ M i ≥ H n − i ( Y, ∂Y ; Q )holds in the case where Y is a compact n -manifold with boundary. To establish this, wechoose first of all a locally flat embedding Y → R ℓ for some ℓ . This can be done by the ONTRYAGIN CLASSES 31 method of [Hi1, Ch.1,Thm.3.4]. In view of this we write Y ⊂ R ℓ . Increasing ℓ if necessary,we may also assume [Hi2] that Y has a normal microbundle in R ℓ , and by [Kis] we may alsoassume that it has a normal bundle N → Y in R ℓ . Choose a neighborhood Y r of Y in R ℓ such that Y is a retract of Y r and Y r is a codimension zero PL submanifold of R ℓ . Accordingto our definition of Λ, we now have to extend the euclidean bundle T Y → Y to a euclideanbundle E r → Y r (which is easy). Then we should replace E r → Y r by E r × R → Y r ,which completes to a disk bundle X → Y r , etc.; we then have to find σ ( X, ∂X ) and pass toPoincar´e duals.— Altogether we now have an embedding Y → X by composing Y incl. −−−−→ Y r incl. −−−−→ E r × R incl. −−−−→ X where the second arrow is any section of the bundle projection E r × R → Y r . To completethe proof, it suffices to show that Y has a trivial normal disk bundle X ′ in X , and to applyproposition 9.3 to the inclusion X ′ → X . Here we note that the existence of a trivial normalbundle (with fibers ∼ = R ℓ +1 ) implies the existence of a trivial normal disk bundle. But it isclear that Y has a normal bundle in X , identified with N × Y ( T Y × R ), and this is clearlytrivial since already N × Y T Y is trivial. (cid:3)
Proposition 9.6.
On vector bundles, the characteristic class Λ agrees with Hirzebruch’stotal L -class.Proof. Let Y be a closed oriented topological manifold of dimension 4 i . By construction,the scalar product h Λ i ( T Y ) , [ Y ] i is equal to the image of σ ( Y ) under the specialization(alias assembly) map H i ( Y ; L • ( Z )) ⊗ Q → π i L • ( Z ) ⊗ Q . In other words it is equal to the signature of Y . This holds in particular when Y is smooth.As this property characterizes the L -class on vector bundles, we have Λ = L on vectorbundles. (cid:3) References [BK]
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School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK
E-mail address : [email protected] Dept. of Math. Sciences, University of Aberdeen, Aberdeen AB24 3UE, Scotland, UK
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