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Intersection Homology Wang Sequence Topology of Stratified Spaces MSRI Publications Volume 58, 2011 Intersection homology Wang sequence FILIPP LEVIKOV ABSTRACT. We prove the existence of a Wang-like sequence for intersection homology. A result is given on vanishing of the middle dimensional inter- section homology group of “generalized Thom spaces”, which naturally occur in the decomposition formula of S. Cappell and J. Shaneson. Based upon this result, consequences for the signature are drawn. For non-Witt spaces X , signature and L-classes are defined via the hyper- i cohomology groups H .X I ICL/, introduced in [Ban02]. A hypercohomology i Wang sequence is deduced, connecting H .I ICL/ of the total space with that of the fibre. Also here, a consequence for the signature under collapsing sphere-singularities is drawn. 1. Introduction The goal of this article is to add to the intersection homology toolkit another useful long exact sequence. In [Wan49], H. C. Wang, calculating the homol- ogy of the total space of a fibre bundle over a sphere, actually proved an exact sequence, which is named after him today. It is a useful tool for dealing with fibre bundles over spheres and it is natural to ask: Is there a Wang sequence for intersection homology? Given an appropriate notion of a stratified fibration, the natural framework for dealing with a question of the kind above would be an intersection homology analogue of a Leray–Serre spectral sequence. Greg Friedman has investigated this and established an appropriate framework in [Fri07]. For a simplified setting of a stratified bundle, however, i.e., a locally trivial bundle over a manifold with a stratified fibre, it seems more natural to explore the hypercohomology spectral sequence directly. In the following we are going to demonstrate this approach. Mathematics Subject Classification: 55N33. Keywords: intersection homology, Wang sequence, signature. 251 252 FILIPP LEVIKOV Section 3 is a kind of foretaste of what is to come. We prove the monodromy case by hand using only elementary intersection homology and apply it to calcu- late the intersection homology groups of neighbourhoods of circle singularities with toric links in a 4-dimensional pseudomanifold. We recall the construction of induced maps in Section 4.1. Because of their central role in the application, the Cappell–Shaneson decomposition formula is explained in Section 4.2. Section 5 contains a proof of the Wang sequence for fibre bundles over simply connected spheres. It is shown that under a certain assumption the middle-dimensional middle perversity intersection homology of generalized Thom spaces of bundles over spheres vanish. The formula of Cappell and Shaneson then implies, that in this situation the signature does not change under the collapsing of the spherical singularities. In Section 6, we demonstrate a second, concise proof — this is merely the sheaf-theoretic combination of the relative long exact sequence and the suspen- sion isomorphism. However, this proof is mimicked in Section 7 to derive a Wang-like sequence for hypercohomology groups H.X I S/ with values in a self-dual perverse sheaf complex S 2 SD.X /. In Section 8, finally, together with Novikov additivity, this enables us to identify situations when collapsing spherical singularities in non-Witt spaces does not change the signature. 2. Basic notions We will work in the framework of [GM83]. In the following X D Xn Xn2 X0 X1 D ? will denote an oriented n-dimensional strati- fied topological pseudomanifold. The intersection homology groups of X with pN respect to perversity pN are denoted by IH i .X /, and the analogous compact- cpN support homology groups by IH i .X /. The indexing convention is also that of [GM83]. Most of the fibre bundles to be considered in the following are going to be stratified bundles in the following sense (see also [Fri07, Definition 5.6]): DEFINITION 2.1. A projection E ! B to a manifold is called a stratified bundle if for each point b 2 B there exist a neighbourhood U B and a stratum- preserving trivialization p1.U / Š U F, where F is a topological stratified pseudomanifold. We will also restrict the automorphism group of F to stratum preserving auto- morphisms and work with the corresponding fibre bundles in the usual sense. Since we will basically need the local triviality, Definition 2.1 is mostly suf- ficient. When we pass to applications for Whitney stratified pseudomanifolds, however, the considered bundles will actually be fibre bundles — this follows from the theory of Whitney stratifications. A stratification of the fibre induces an obvious stratification of the total space with the same l-codimensional links, INTERSECTIONHOMOLOGYWANGSEQUENCE 253 namely by EkCnl — the total spaces of bundles with fibre Fkl and n the dimension of B. 3. Mapping torus PROPOSITION 3.1 (INTERSECTION HOMOLOGY WANG SEQUENCE FOR S 1). Let F D Fn Fn2 F0 be a topological stratified pseudomanifold, W F ! F a stratum and codimension preserving automorphism, i.e., a stratum preserving homeomorphism with stratum preserving inverse such that both maps respect the codimension. Let M be the mapping torus of , i.e., the quotient space F I=.y; 1/ s ..y/; 0/. Denote by i W F D F 0 Œ M the inclusion. Then the sequence cpN id cpN i cpN @ cpN IH k .F/ ! IH k .F/ IH k .M/ IH k1.F/ is exact. PROOF. The proof is analogous to the one for ordinary homology. Start with the quotient map q W .F I; F/ ! .M; F/ and look at the corresponding diagram of long exact sequences of pairs. The boundary of F I is a codimension 1 stratum and hence not a pseudomanifold. We have either to introduce the notion of a pseudomanifold with boundary here or work with intersection homology for cs-sets [Kin85; HS91]. However, we can also manage with a work-around: Define I" WD ."; 1 C "/; @I" WD .";"/ [ .1 "; 1 C "/; F" WD F ."; "/: We extend the identification .y; 1/ s ..y/; 0/ to F I" by introducing the quotient map q W F I" ! M, .1.y/; 1 C t/ if t2 ."; 0 q.y; t/ D 8.y; t/ if t2 .0; 1/ <..y/; t 1/ if t2 Œ1; 1 C "/: Evidently, M D q.F I"/.: Now F @I" D .F @I"/nC1 .F @I"/ .F @I"/0 is an open subpseudomanifold of F I" and F D Fn Fn2 F0 sits normally nonsingular in M. Hence the inclusions induce morphisms on intersection homology and we get a morphism of the corresponding exact sequences of pairs: 0 ... cpN @ ... cpN j ... cpN 0 ... ::: .................................................................. ............................................................................................... .............................................................................. ............................................................. ::: IH k .F I"; F @I"/ IH k1.F @I"/ IH k1.F I"/ . q . q . q . ....... ....... ....... ..... ..... ..... ... cpN @ ... cpN i ... cpN ... ::: ................................................................................................ .................................................................................................................................................. .......................................................................................................... .................................................................... ::: IH k .M ; F"/ IH k1.F"/ IH k1.M / 254 FILIPP LEVIKOV The “boundary” of F I" is the disjoint union of two components of the form F R, so j is surjective and the outer arrows are zero maps. The connecting morphism @ is injective and therefore an isomorphism onto its image, i.e., onto ker j cpN cpN D .˛;ˇ/ j ˛ 2 IH k .F."; C"//; ˇ 2 IH k .F.1"; 1C"//; Œ˛Cˇ D 0 ˚ cpN R cpN « Df.˛; ˛/gŠ IH k .F / Š IH k .F/: cpN cpN The middle q maps .˛; ˛/ to .˛ .˛// 2 IH k .F"/ Š IH k .F/. Since q commutes with @, one has @ ı q ı @j1 D q j cpN D id : ker jŠIH k .F / Hence, we have the diagram q j ... cpN ker j ... cpN i... cpN ... cpN : : : .......................... ...................................................................................................................................................................... ........................................... IH .M / .................................... IH .M ; F / IH k .F/ IH k .F/ k k " . ........ ko . ker j . @ . 1 . ....... o..... @j . cpN q... cpN ...................................... IH kC1.F I"; F @I"/ IH kC1.M; F"/ where the top sequence is exact and the bottom square is commutative. As in ord- cpN cpN inary homology one can show that q W IH .F I"; F @I"/ ! IH .M; F"/ k cpNk is an isomorphism. Finally, observe that on the right hand side IH k .M; F"/Š cpN 1 ˜ IH k1.F/ via @ ı q . Let X D X4 X1 X0 be a compact stratified pseudomanifold with X0 D ?. Then, the stratum of codimension 3 is just a disjoint union of circles X1 D 1 1 S tt S . If we assume X to be PL, the link L at a point p 2 X1 is independent of p within a connective component of X1. Furthermore, in X4, there is a neighbourhood U of the circle containing p, which is a fibre bundle 1 over S and hence homeomorphic to the mapping torus M with Š W c˚.L/ ! c˚.L/: Putting this data together and using the Wang sequence we can compute cpN cpN IH k .U /, with U a neighbourhood of X1 X4. The group IH k .X / can then be computed via the Mayer–Vietoris sequence. In this section we restrict ourselves to the case of L being a torus T 2. While the orientation preserving mapping class group of the torus is known to be SL.2I Z/, we have to make the following restriction on its cone: In the following, we look only at those automorphisms W c˚.T 2/ ! c˚.T 2/ which are induced INTERSECTIONHOMOLOGYWANGSEQUENCE 255 Š by an automorphism of the underlying torus W T 2 ! T 2.1 It is given by a matrix ˛ 2 SL.2I Z/ and by abuse of notation we will again write ˛ for this torus automorphism. ˚ cpN 2 id .c.˛// cpN 2 Defining k to be the map IH k .c˚.T //!IH k .c˚.T //, we obtain the sequence cpN 2 k IH k .c˚.T // cpN 2 i cpN @ cpN 2 IH k .c˚.T // IH k .M˛/ IH k1.c˚.T // For the open cone we have cpN 2 IH 0 .T / for k D 0; cpN 2 cpN 2 N IH k .c˚.T // D 8IH 1 .T / for pN D 0 and k D 1; <0 else: cpN : Clearly, IH k .M˛/ D 0 for k 3.
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