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 intersection 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 homology 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 suspension 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 stratified 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 automorphisms and work with the corresponding fibre bundles in the usual sense. Since we will basically need the local triviality, Definition 2.1 is mostly sufficient. 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. Now examine the nontrivial part of the sequence

cpN @ cpN 2 1 cpN 2 i 0 IH 2 .M˛/ IH 1 .c˚.T // IH 1 .c˚.T // cpN @ cpN 2 0 cpN 2 i cpN IH 1 .M˛/ IH 0 .c˚.T // IH 0 .c˚.T // IH 0 .M˛/ 0:

Since c˚.˛/0 maps a point to a point, clearly c˚.˛/0 D id, hence 0 D id id D 0. cpN Z It follows that IH 0 .M˛/ D . Note that the only possible perversities in this example are 0N and tN. So far ctN 2 we have not distinguished between them. Due to IH 1 .c˚.T // D 0 there is ctN ctN Z IH 2 .M˛/ D 0 and IH 1 .M˛/ D . For the zero perversity, we have

c0N @ c0N 2 1 c0N 2 c0N 0 IH 2 .M˛/ ! IH 1 .c˚.T //! IH 1 .c˚.T // IH 1 .M˛/ ko Z ˚ Z c0N 2 IH 0 .c˚.T // 0 c0N 2 2 The group IH 1 .c˚.T // is isomorphic to H1.T / and is therefore generated by the corresponding homology classes of the torus. Hence, .c˚.˛// is just c0N Z Z the matrix ˛. If ˛ D id; 1 D 0 and IH 2 .M˛/ Š ˚ . In the general c0N case, IH 2 .M˛/ is isomorphic to im @ D ker 1. We examine the determinant

1I believe that in the PL context this does not constitute a real restriction. With [Hud69, Theorem 3.6C] we can ﬁnd an admissible triangulation of c˚.T 2/, such that becomes simplicial. Furthermore, it is not difﬁcult to see, that the simplicial link of the cone point L.c/ is preserved under . Since the geometric realization of L.c/ is a torus, we get a candidate for . By linearity, every slice between L.c/ and the cone point c is mapped by . If we could extend the argument to the rest of c˚.T 2/ the goal would be achieved. 256 FILIPP LEVIKOV of the matrix: det 1 D det.id ˛/ D p˛.1/, where p˛.t/ is the characteristic polynomial of ˛, which is

a11 a12 p˛.t/ D det t id a a D .t a11/.t a22/ a21a12 21 22 2 D t .a11 C a22/t C .a11a22 a21a12/ D t 2 tr ˛ t C det ˛ D t 2 tr ˛ t C 1:

Here, tr ˛ is the trace of ˛ 2 SL.2I Z/. Thus we have det 1 D 2 tr ˛ and get Z ˚ Z if ˛ D id; c0N Z IH 2 .M˛/ Š 8 if tr ˛ D 2 and ˛ ¤ id; <0 if tr ˛ ¤ 2: c0N 2 : Since IH 0 .c˚.T // is free, the sequence above reduces to a split short exact sequence c0N c0N 2 0 coker 1 IH 1 .M/ IH 0 .c˚.T // 0: c0N Z Hence IH 1 .M/ Š ˚ coker 1. In this ﬁnal case our interest reduces to a cokernel calculation of the 2 2-matrix 1 D id ˛. The image im 1 Z ˚ Z is of the form nZ ˚ mZ; n; m 2 Z and so every group Z ˚ Z=nZ ˚ Z=mZ can c0N be realized as IH 1 .M/. In particular a torsion intersection homology group may appear. Using det 1 D det.id ˛/ D 2tr ˛ as above, we immediately see that Z ˚ Z if ˛ D id; coker 1 Š 0 if tr ˛ D 1; 3: Summarizing all these results we get: 2 PROPOSITION 3.2. Let M˛ be the mapping torus over the open cone c˚.T / of Š a torus glued via ˛ W T 2 ! T 2. Then its intersection homology groups are

ctN Z if k D 0; 1; IH .M˛/Š k 0 if k 2;

Z if k D 0; 8Z ˚ Z ˚ Z if k D 1 and ˛ D id ˆZ ˆ if k D 1 and tr ˛ D 1; 3; ˆZ c0N ˆ ˚ coker.id ˛/ if k D 1 (in general); IH k .M˛/ŠˆZ <ˆ if k D 2; tr ˛ D 2 and ˛ ¤ id; Z ˚ Z if k D 2 and ˛ D id; ˆ ˆ0 if k D 2 and tr ˛ ¤ 2; ˆ ˆ0 if k 3: ˆ :ˆ INTERSECTIONHOMOLOGYWANGSEQUENCE 257

EXAMPLE 3.3. Let X be the fibre bundle over S 1 with fibre ˙T 2 and monodromy ˛ 2 SL2.Z/. We assume ˙˛ to be orientation preserving, i.e., the sus- 1 1 pension points are fixed under it. The space X has a filtration X4 X1 DS tS 2 1 and our situation applies. Let ˛ be 1 0 . First, using the ordinary Wang sequence, we compute the homology of the total space E of the fibre bundle T 2 ! E ! S 1 with the same monodromy:

@ 2 id ˛ 2 @ 2 0 ! H3.E/ ! H2.T / ! H2.T / ! H2.E/ ! H1.T / id ˛ 2 @ 2 id ˛ 2 ! H1.T / ! H1.E/ ! H0.T / ! H0.T / ! H0.E/ ! 0:

In degrees 2 and 0, the map ˛ is the identity, so we substitute zeros for id ˛ to see that H3.E/ Š Z Š H0.E/. In degree 1, the map ˛ is just the matrix ˛. Using im @2 D ker.id ˛/ Š Z, we get the sequence

2 @ 0 ! H2.T / ! H2.E/ ! Z ! 0; which yields H2.E/ Š Z ˚ Z and

0 ! coker.id ˛/ ! H1.E/ ! Z ! 0:

It follows by coker.id ˛/ Š Z that H1.E/ Š Z ˚ Z. Let us now compute the intersection homology groups of X via the Mayer–Vietoris sequence. The neighbourhoods of the two S 1 are of the desired form, i.e., mapping tori over c˚.T 2/ and their intersection is a ﬁbre bundle over S 1 with ﬁbre T 2 R, so that the intersection homology groups are just H.E/ from above. Looking at the exact sequence cpN cpN cpN cpN 0 IH 4 .X / ! IH 3 .E/ !IH 3 .M˛/˚IH 3 .M˛/! ; jj jj 0 0 cpN Z we see that IH 4 .X / Š H3.E/ Š . In degree 0 the inclusion of E induces an injection on homology, i.e.,

cpN cpN cpN 0 H0.E/ ! IH 0 .M˛/ ˚ IH 0 .M˛/ ! IH 0 .X / ! 0; cpN Z and IH 0 .X / Š as it should be. Turning to the interesting degrees, we look at ctN ctN pN D .0; 1;:::/ ﬁrst. Due to IH 2 .M˛/ D 0, there is IH 2 .X / Š ker .i ˚ i/1. ctN Finally IH 1 .X / is isomorphic to the cokernel of the inclusion .i ˚ i/1 W ctN ctN Z Z H1.E/ ! IH 1 .M˛/˚IH 1 .M˛/ Š ˚ , which is the diagonal map; hence ctN Z ctN Z IH 1 .X / Š and IH 2 .X / Š ker .i ˚i/1 Š . Similarly, for pN D .0; 0;:::/ c0N Z c0N Z Z we have IH 3 .X / Š ker .i ˚ i/2 Š ; with IH 1 .M˛/ Š ˚ it follows c0N Z Z c0N Z IH 1 .X / Š ˚ . And IH 2 .X / Š coker .i ˚ i/2 Š . Because all the 258 FILIPP LEVIKOV groups are free, the duality is already seen working with integral coefﬁcients, especially c0N ctN Z Z IH 3 .X / Š IH 1 .X / Š ˚ ; c0N ctN Z IH 1 .X / Š IH 3 .X / Š :

4. Some more advanced tools 4A. Normally nonsingular maps. Intersection homology is not a functor on the full subcategory of Top consisting of pseudomanifolds, since induced maps do not exist in general. However, on the category of topological pseudomanifolds and normally nonsingular maps, intersection homology is a bivariant theory in the sense of [FM81]. This fact is often suppressed. Since most of the maps which we will encounter are normally nonsingular, we recall in this section how induced maps are constructed. See particularly [GM83, 5.4].

DEFINITION 4.1. A map f W Y ! X between two pseudomanifolds is called normally nonsingular (nns) of relative dimension c D c1 c2 if it is a composition of a nns inclusion of dimension c1 — meanining that Y is sitting in a c1-dimensional tubular neighbourhood in the target — and a nns projection, i.e., a bundle projection with c2-dimensional manifold ﬁbre.

EXAMPLE 4.2. An open inclusion U Œ X is normally nonsingular. The inclusion of the ﬁbre F D b F Œ E, where E is ﬁbred over a manifold is normally nonsingular. The projection Rn X ! X is normally nonsingular.

PROPOSITION 4.3 [GM83, 5.4.1, 5.4.2]. Let f W Y ! X be normally nonsingular of codimension c. Then there are isomorphisms

! f ICpN .X / Š ICpN .Y /Œc and f ICpN .X / Š ICpN .Y /: DEFINITION 4.4. If f W Y ! X is a proper normally nonsingular map of codimension c, we have induced homomorphisms pN pN pN pN f W IH k .Y / ! IH k .X / and f W IH k .X / ! IH kc.Y /: They are constructed by considering the adjunction morphisms of the adjoint ! pairs .Rf!;f / and .f ; Rf/,

! Rf!f ICpN .X / ! ICpN .X / and ICpN .X / ! Rff ICpN .X /; by combining them with the proposition above and by ﬁnally applying hypercohomology. We will also need induced maps on intersection homology with compact supports, which are not discussed in [GM83]. The above construction of Goresky cpN and MacPherson works equally well for IH . If f is not proper, the map INTERSECTIONHOMOLOGYWANGSEQUENCE 259

cpN cpN f still exists. For the case of compact supports, f! W IH k .Y / ! IH k .X / can be constructed in the same manner. These different maps are listed in the following table — note that only f and f for proper f are explicitly men- tioned in [GM83, 5.4].

f proper f not proper pN pN f W IH k .Y / ! IH k .X / cpN cpN cpN cpN f! W IH k .Y / ! IH k .X / f! W IH k .Y / ! IH k .X / pN pN pN pN f W IH k .X / ! IH kc.Y / f W IH k .X / ! IH kc.Y / ! cpN cpN f W IH k .X / ! IH kc.Y / 4B. Behaviour under stratified maps. Computing intersection homology invariants of one space out of the invariants of the other often relies on the decomposition formula of S. Cappell and J. Shaneson [CS91]. Since we will need it in the application below, we briefly recall it in this section. Let f W X n ! Y m be a stratified map between closed, oriented Whitney stratified sets of even relative dimension 2t D n m, Y having only even- b codimensional strata. Let S 2 Dc .X / be a self-dual complex. Denote by V the 2 set of components of pure strata of Y . For each y 2 Vy 2 V, define

1 1 Ey WD f .c L.y// [f 1L.y/ c f .L.y//; where L.y/ is the link of the stratum component Vy containing y. If y lies in 1 the top stratum, we set Ey D f .y/. We have the inclusions

i y 1 y Ey ‹ f .N˚ .y// Œ X; where N.y/ is the normal slice of y. Note that N.y/ Š c L and @N.y/ Š L.y/ (see [GM88] or [Ban07, 6.2]). Deﬁne now the complex

cone ! S .y/ D ct1RiyyS ;

cone 3 1 where stands for truncation over the cone point of c f .L.y// and 2c D 2c.V / D n dim V is the codimension of V .

2Here, c L stands for the closed cone L Œ0; 1=L f0g. 3There is a general notion of truncation over a closed subset in [GM83, 1.14]. Let c be the cone-point. b For A 2 Dc .Ey /, the derived stalks are

i cone 0 if x D c and i > p; H .p A /x D i H .A /x otherwise. 260 FILIPP LEVIKOV

V SV SV For V 2 , let f be the local coefﬁcient system over V with stalk . f /z D ct H .EzI S .z//. There is an induced nondegenerate bilinear pairing SV SV R z W . f /z . f /z ! : ! If S is the intersection chain complex ICmN .X /, the pull-back y ICmN .X / is clearly ICmN .Ey nfcg/. Because of the stalk vanishing of ICmN .Ey nfcg/, the cone truncation ct1 is the usual truncation ct1 and hence S .y/ D ct1Riy ICmN .Ey nfcg/; which is simply the Deligne extension ICmN .Ey/ of ICmN .Ey nfcg/ to the point. N SV Denoting by ICmN .V I f / the lower-middle perversity intersection chain com- SV plex on the closure of V with coefﬁcients in the local system f , we can now formulate the important decomposition formula of Cappell and Shaneson.

THEOREM 4.5 [CS91, Theorem 4.2]. There is an orthogonal decomposition up to algebraic bordism of self-dual complexes of sheaves N SV RfS Œt j ICmN .V ; f /Œc.V /; VM2V where j W VN Œ Y is the inclusion. We abstain from giving the deﬁnition of algebraic bordism here and refer to the original paper or to Chapter 8 of [Ban07]. All we need for the application is the following, where for a self-dual sheaf S over X , .X; S/ denotes the signature of the pairing on the middle-dimensional hypercohomology induced by self-duality.

PROPOSITION 4.6. If two self-dual complexes over X , S1 and S2 are (algebraically) bordant, then .X; S1/ D .X; S2/. PROOF. See [Ban07, Cor. 8.2.5], for example. ˜

PROPOSITION 4.7 [CS91, 5.5]. If , in the setting above, Li.X; A / denotes the i-th L-class of the self-dual sheaf A over a pseudomanifold X , we have

Li.Y; RfS Œt/ D fLi.X; S /: N SV N N SV THEOREM 4.8. With the notation Li.V ; f / for Li.V ; ICmN .V I f // we get N SV fLi.X; S / D jLi.V ; f /: VX2V and bearing in mind that .X / D "L0.X /, where " is the augmentation, we conclude: N SV COROLLARY 4.9. .X; S / D .Y; RfS Œt/ D .V ; f /: VX2V INTERSECTIONHOMOLOGYWANGSEQUENCE 261

In the case of simply connected components of Y , all the coefﬁcient systems become constant and using multiplicativity formulae [Ban07, 8.2.19, 8.2.20], we get:

THEOREM 4.10. Assume each V 2 V to be simply connected and choose a basepoint yV for every V 2 V. Then N fLi.X / D .EyV /jLi.V /: VX2V And ﬁnally, for the signature: N COROLLARY 4.11. .X / D .EyV /.V /: VX2V

5. The general simply connected case

PROPOSITION 5.1 (WANG SEQUENCE FOR n 2). Let F E ! S n be a stratiﬁed bundle (2.1) with F a topological pseudomanifold with ﬁnitely generated cohomology, n 2. Let j W F Œ E be the inclusion. (i) For intersection homology the sequence

pN j pN pN j pN IH k .E/ ! IH kn.F/ IH k1.F/ ! IH k1.E/ is exact. (ii) For intersection homology with compact supports the sequence

cpN j cpN cpN j cpN IH k .E/ ! IH kn.F/ IH k1.F/ ! IH k1.E/ is exact. (iii) These sequences are natural with respect to fibre-preserving proper normally nonsingular maps between stratified bundles over S n, i.e., let F 0 ! E0 ! S n be another fibre bundle such that there is a commutative triangle f ... E0 ...... E ...... 0 ...... ...... S n

with f proper normally nonsingular, then there is a commutative diagram of the corresponding Wang-sequences induced by f — both in a covariant and a contravariant way. 262 FILIPP LEVIKOV

PROOF. (i) We begin with the hypercohomology spectral sequence ([Bry93]) HpCq n pN for A WD R ICpN .E/, which converges to .S ; A / Š IH pq.E/. Let U S n be an open set such that 1.U / Š U F, then by 4.3

ICpN .E/j1.U / Š ICpN .U F/ Š pr ICpN .F/Œn: By IV.7.3 of [Bre97], the sheaf Hq.A/, being the Leray sheaf of the ﬁbration, is locally constant. Hence, by the assumption n 2, it is constant with stalk Hq pN .F; ICpN .F/Œn/ D IH qn.F/: Finally pN p;q IH .F/ if p D 0 or p D n; E Š qn 2 0 else.

Hence E2 Š : : : Š En and the sequence collapses at n C 1. Now, the proof can be ﬁnished as in the ordinary case (see [Spa66, 8.5], for instance). In order to pN pN show that IH k .F/ ! IH k .E/ is induced by the inclusion j W F D b0 F Œ E, look at the ﬁbration 0 F ! b0 F ! b0 n for b0 2 S the north pole. We have a commutative diagram 0 ...... ...... b0 F... b0 ...... j . j . . 0 ...... ... E ...... S n :

0 For R.j0 / ICpN .b0 F/ there is a corresponding spectral sequence converg- ing to HpCq pN .b0 F; ICpN .b0 F// Š IH pq.F/: If we start with ! RRjj ICpN .E/ ! R ICpN .E/ and use the commutative square above, we get a morphism

0 R.j0 / ICpN .b0 F/ ! R ICpN .E/: This induces pN pN IH i .F/ ! IH i .E/; which is j by construction (cf. 4.3 and 4.4). The E2-term of the spectral 0 sequence associated to R.j0 / ICpN .b0 F/ is

p;q p n q 0 E2 D H .S ; H .R.j0 / ICpN .b0 F///: INTERSECTIONHOMOLOGYWANGSEQUENCE 263

Since here j0 is just extension by zero the group on the right is isomorphic to p pN n;q nCr;qrC1 H .b0; IH q.F//: For both sequences, the differentials Er ! Er n;q n;q are zero for all r 2, and so we have epimorphisms Er “ E1 . Finally by the commutative diagram (denoting by 0 the terms of the spectral sequence 0 associated to R.j0 / ICpN .b0 F/)

...... q n;in ...... n;in ...... H .E; IC .E// En E1 pN ...... j . ko . . ...... 0 0 ...... Š...... Š ...... q n;in ...... n;in ...... H .F; IC .F// En E1 pN we deduce that the upper composition is j, as stated. A very similar argument pN pN works for IH k .E/ ! IH kn.F/. (ii) Consider the hypercohomology spectral sequence for the complex B WD R! ICpN .E/. The main argument is as before. The spectral sequence converges pCq n cpN th to H .S ; B / D IH pq.E/. Being the q derived functor of !, the stalk q q cpN of the Leray sheaf H .B / is Hc .F; ICpN .E// Š IH qn.F/ (see [Bor84, VI, 2.7], for instance). Hence the E2-terms are: 0;q cpN E2 Š IH qn.F/; n;qnC1 cpN E2 Š IH q1.F/: These yield the second sequence. The proof that the maps involved in this sequence are j and j follows as in (i). (iii) Again, we use the fact that the hypercohomology spectral sequence is natural with respect to morphisms of sheaves over the base space. For the covariant case, we have to construct

0 0 0 0 f W R ICpN .E / ! R ICpN .E/ or f W R! ICpN .E / ! R! ICpN .E/; as the case may be producing morphisms between the terms of the Wang sequences. Then the corresponding maps will commute. Take the adjunction morphism ! Rf!f ICpN .E/ ! ICpN .E/; cpN apply R and use functoriality. The case of IH is analogous. Since f is proper, we have Rf! D Rf. Observe that when working with intersection homology with compact supports f need not be proper4! In the contravariant case, we proceed as above, using the other adjunction morphism ˜ ICpN .E/ ! Rff ICpN .E/:

4See also the comment at the end of Section 4A. 264 FILIPP LEVIKOV

Now we are going to use this sequence in a concrete computation. PROPOSITION 5.2. Let F k ! E ! S n be a locally trivial fibre bundle with F a topological pseudomanifold, n 2; n C k C 1 even. Define M WD cE [E EN , where EN is the total space of the induced — meaning that the structure group acts levelwise — fibre bundle

c ./ cF ! EN ! S n: Suppose further that the following condition (S) is fulﬁlled for the Wang sequence of E: cmN “ cmN (S) the map j W IH .nCkC1/=2.E/ IH .nCkC1/=2.F/ is surjective. Then cmN IH .nCkC1/=2.M / D 0: PROOF. Throughout the proof, the perversity shall be the lower middle perversity mN , unless stated otherwise. Assume for now that n D 2b, k D 2a 1, with a; b 1; and with the cone formula there holds IH c.F/ if i < a; IH c.cF˚ / Š i i 0 if i a:

c˚./ Using the Wang sequence of Proposition 5.1 for cF˚ ! EN˚ ! S n, c c N˚ c c ! IH aCb.cF˚ / ! IH aCb.E/ ! IH ab.cF˚ / ! IH aCb1.cF˚ / ! we get c N˚ c IH aCb.E/ Š IH ab.F/: For the cone on E, there is: IH c.E/if i < a C b; IH c.cE˚ / Š i i 0 if i a C b: Now consider the Mayer–Vietoris sequence5

i c aCb c c N˚ c ! IH aCb.E/ ! IH aCb.cE˚ / ˚ IH aCb.E/ ! IH aCb.M / ! which reduces to i aCb c c ! IH ab.F/ IH aCb.M / i c aCb1 c c N˚ IH aCb1.E/ ! IH aCb1.E/ ˚ IH aCb1.E/ ˚ 5To avoid pseudomanifolds with boundary, we take the open part EN of the induced bundle EN in the Mayer–Vietoris decomposition. INTERSECTIONHOMOLOGYWANGSEQUENCE 265

The map iaCb1 is easily seen to be injective and due to (S), iaCb is surjective. Finally, let n D 2b C 1; k D 2a; a; b 1. By the cone formula we have: IH c.E/ i < a C b C 1 IH c.cE˚ / Š i i 0 i a C b C 1 and IH c.F/ i < a C 1 IH c.cF˚ / Š i i 0 i a C 1: Similarly, the Wang sequence yields c N˚ c IH aCbC1.E/ Š IH ab.F/: Now, as above the Mayer–Vietoris sequence gives i i aCbC1 c c c aCb1 c c N˚ ! IH ab.F/ ! IH aCbC1.M / ! IH aCb.E/ ! IH aCb.E/˚IH aCb.E/ c where ker iaCb D 0 and im iaCbC1 D IH ab.F/ due to (S). Hence, c ˜ IH aCbC1.M / D 0: COROLLARY 5.3. If M is a Witt space and n C k C 1 is divisible by 4, the signature .M / vanishes (of course it always vanishes if nCkC1 is not divisible by 4.) Let us now formulate an important consequence of the observations above:

PROPOSITION 5.4. Let X be a Whitney stratified Witt space of dimension 4k, with a disjoint union of spheres as the singular locus ˙ D S n1 tt S nl . Assume nj 2 for 1 j l. Let Y be the space obtained from X by collapsing n the spheres S j to points yj and let f W X ! Y be the collapsing map. Given n a fibre bundle neighbourhood of S j , we denote by Ej the corresponding fibre n bundle with fibre the link of S j . If for all 1 j l, Ej satisfies (S), the signature of X does not change under f , i.e., .X / D .Y /:

PROOF. Because of 4.11, we have N .X / D .EyV /.V / VX2V where the sum is taken over all strata. When we isolate the contribution of the top stratum, this looks like N D .Eyj V /.V / C .Y /: Xyj 266 FILIPP LEVIKOV

The Eyj V , in turn, are of the form M of Proposition 5.2 and since the underlying ﬁbrations satisfy (S), the resulting signatures vanish by 5.3. ˜ Similarly for the L-classes we have:

PROPOSITION 5.5. In the situation above,

fL.X / D L.Y /: The following two examples show, that the introduced condition (S) is indeed fulﬁlled for certain ﬁbre bundles.

EXAMPLE 5.6. In the setting above let the base sphere be of odd dimension n D 2b C 1. If we are interested in computing the signature of E, its dimension has to be divisible by 4 — otherwise it is trivial anyway. In this case the ﬁbre F has even dimension k D 2a, so that .k n C 1/=2 is odd. Thus, the vanishing of odd dimensional intersection homology of F would imply (S). See [Roy87] for examples of spaces, for which the intersection homology vanishes in odd degrees.

EXAMPLE 5.7. Let the dimension of the sphere be greater than the dimension of the fibre plus 1, i.e., k C 1 < n. Then .k n C 1/=2 is negative and the corresponding homology group is zero, thereby (S) is fulfilled. Since we have not studied the intersection pairing on M , the condition (S) is clearly only sufficient and not necessary. However, the following ”counterex- ample” to the proposition is a case where (S) does not hold.

EXAMPLE 5.8. Let X be CP 2 stratiﬁed as CP 2 CP 1 DS 2 and f be the map, collapsing the 2-sphere to a point. So the target is Y D S 4 ŒS 2. Obviously, .X / ¤ .Y /. The link of CP 1 is a circle and the bundle we have to check (S) for is the Hopf bundle S 3 ! S 2. However 3 2 H2.S / ! H0.S / is not onto and (S) fails.

6. A new proof

The application to the signature in the last section suggests a similar approach in the setting of spaces which no longer satisfy the Witt condition, however still posses a signature and L-classes. The suitable homology groups for deﬁn- i ing these invariants are the hypercohomology groups H .I ICL/ of Banagl [Ban02]. In the next section we will establish a Wang-like exact sequence for these groups i.e., for hypercohomology with values in a self-dual sheaf complex arising from a Lagrangian structure along the odd-codimension strata. Compare INTERSECTIONHOMOLOGYWANGSEQUENCE 267 also [Ban07] for a concise exposition. The proof will be modeled on another elegant proof of the Wang sequence without the usage of the spectral sequence. This will be demonstrated in the following. Suspension isomorphisms in intersection homology are very familiar. For the functoriality, however, we would like to have explicit maps realizing these isomorphisms:

LEMMA 6.1 (SUSPENSION ISOMORPHISM). Let F be a pseudomanifold. The inclusion l W F D 0 F Œ Rn F induces isomorphisms pN Rn pN (a) l W IH k . F/ ! IH kn.F/, cpN cpN Rn (b) l! W IH k .F/ ! IH k . F/. PROOF. (a) Let p W Rn F ! F be the normally nonsingular projection. By [Bor84, V,3.13] Rp ı p ' id, so the adjunction morphism is an isomorphism

Š Rn ICpN .F/ ! Rpp ICpN .F/ Š Rp ICpN . F/Œn:

Applying hypercohomology we get

pN Š pN Rn p W IH k .F/ ! IH kCn. F/: Now p ıl D id and hence l ıp ' id. Thereby, l is the inverse of p and the statement follows.

b (b) is similar to (a), but uses the fact that DX DX A Š A for A 2 Dc .X /, and the duality between p and p!. ˜ PROPOSITION 6.2. Let F E ! S n be a stratiﬁed bundle with F a topological pseudomanifold. Denote by

j W F D b0 F Œ E, i W E n b0 F D U F Œ E, k W b1 F Œ E the inclusions, where b0 is the north pole and b1 the south pole. Then there are the following long exact sequences:

pN j pN k pN pN IH k .F/ ! IH k .E/ ! IH kn.F/ IH k1.F/ cpN k! cpN j pN cpN IH k .F/ ! IH k .E/ ! IH kn.F/ IH k1.F/ :

PROOF. In the following, trivializations of the ﬁbre bundle E are always in- Š volved. However, for every pseudomanifold X , h W X ! X implies ICpN .X / Š h ICpN .X / Š h ICpN .X /. Therefore, for the proof we can suppress them. 268 FILIPP LEVIKOV

We begin with the distinguished triangle

.. ! ...... Rjj ICpN .E/ . ICpN .E/ ...... Œ1 ...... Rii ICpN .E/

! and keeping in mind that j ICpN .E/ Š ICpN .F/, i ICpN .E/ Š ICpN .U F/ we apply hypercohomology to get

pN j pN i pN ! IH k .F/ ! IH k .E/ ! IH k .U F/ ! : pN The third term is isomorphic to IH kn.F/ under l by the preceding lemma. However by the commutative triangle

...... k ...... b1 F ... E ...... l ...... i ..... U F we have k ' l ı i and the sequence for closed supports is proven. Now turn to the case of compact supports.6 Consider the triangle

...... Ri!i ICpN .E/ ... ICpN .E/ ...... Œ1 ...... Rjj ICpN .E/ and apply hypercohomology with compact supports to get

i j Hk ! cpN Hk ! c .EIRi! ICpN .UF//!IH k .E/! c .EIRj ICpN .F/Œn/! :

Now Rj D Rj! as j is a closed inclusion. Hence Hk Hk cpN c .EI Rj ICpN .F/Œn/ Š c .FI ICpN .F/Œn/ Š IH kn.F/: For the ﬁrst term, we have

k k l! cpN H .EI Ri! IC .U F// Š H .U FI IC .U F// IH .F/ c pN c pN Š k and with i! ı l! D k! the assertion follows. ˜

6 b 1 Recall that for f W X ! Y , A 2 Dc .X / and Z X , we have c .Z;fA / © c .f .Z/; A /. 1 However c .Z;f!A / Š c .f .Z/; A /. INTERSECTIONHOMOLOGYWANGSEQUENCE 269

7. The non-Witt case

7A. The category SD(X ). Originally, Goresky and MacPherson defined the signature for spaces with only even-codimensional strata. In [Sie83], Siegel mN nN generalizes the definition to Witt spaces. If IH .X / © IH .X /, there still is a method to define a signature and L-classes for a pseudomanifold X compatible with the old definition. In his work [Ban02], Banagl establishes a corresponding framework and decomposition results similar to those of Cappell and Shaneson are presented in further papers. In this section we merely give the definition.

DEFINITION 7.1. Let X D Xn X0 be an oriented pseudomanifold with orientation Š o W D ! R Œn: U2 U2

For k 2, we write Uk WD X nXnk. Deﬁne SD.X / as the full subcategory of b b Dc .X / of those S 2 Dc .X / satisfying the following: Š R (SD1) Normalization: There is an isomorphism W U2 Œn ! S jU2 . (SD2) Lower bound: Hi.S/ D 0, for i < n. i (SD3) Stalk condition for nN: H .S jUkC1 / D 0, for i > nN.k/ n; k 2. (SD4) Self-duality: There is an isomorphism d W DX S Œn ! S compatible with the orientation, i.e., such that the square

... R ...... U2 Œn S jU2 ... Š ...... o . . . Š Š . djU . . 2 ...... D 1 . X Œn ... D ...... D S j Œn commutes. U2 Š X U2

We refer to [Ban02] for results on this category, especially for the structure theorem, establishing the relation between S 2 SD.X / and a choice of La- grangian structures along odd-codimensional strata of X .

REMARK 7.2. If X is a Witt space, SD.X / consists up to isomorphism only of C 2 ? ICmN .X /. On the other hand, SD.X / might be empty — e.g., SD.˙ P / D .

THEOREM 7.3 [Ban02, Theorem.2.2]. For S 2 SD.X /, there is a factorization

˛ ˇ ICmN .X / ! S ! ICnN .X /; 270 FILIPP LEVIKOV that is compatible with the normalization (and is unique with respect to this property) and such that

...... ˛ ...... ICmN .X / .. S ...... Š . Š . . . d ...... D . X ˇŒn ... D ...... X ICnN .X / DX S commutes.

Thus, an object in SD.X / is in fact a self-dual interpolation between ICmN .X / and ICnN .X /. It is obvious that in the case of X being a Witt space, SD.X / consists (up to quasi-isomorphism) only of ICmN .X /. DEFINITION 7.4. Let X n be a closed stratiﬁed topological pseudomanifold, not necessarily Witt and S 2SD.X /. In case n is divisible by 4, deﬁne .X n; S; d/ to be the signature on Hn=2.X n; S/ induced by the self-duality of S.

REMARK 7.5. If X happens to be a Witt space, .X n; S; d/ is the usual intersection homology signature due to Theorem 7.3. Finally, in order to speak of the signature of a pseudomanifold (as long as SD.X / ¤ ?), we need the following important result:

THEOREM 7.6 [Ban06, 4.1]. Let X n be an even-dimensional closed oriented ? pseudomanifold with SD.X / ¤ . For .S1; d1/;.S2; d2/ 2 SD.X / one has

n n .X ; S1; d1/ D .X ; S2; d2/: 7B. Hypercohomology Wang sequence. Before we deduce the exact sequence for hypercohomology with values in SD-sheaves, we have to determine what the involved complexes of sheaves are going to be. Starting with a SD complex over the total space E we deﬁne a SD complex over the ﬁbre F in a canonical way. We will need the following little lemma.

LEMMA 7.7. For the inclusion j WX n0ŒX nRm with associated projection p W X n Rm ! X n we have j ! ' j Œm and thereby p! ı j ! ' id :

PROOF. By [Ban02, Lemma 5.2], p ı j ' id. Consequently, using pŒm ' p!([Ban02, Lemma 4.2, Proof]), we get j ! ' j ! ı p ı j ' j ! ı p! ı j Œm ' .p ı j /! ı j Œm ' j Œm: INTERSECTIONHOMOLOGYWANGSEQUENCE 271

The second identity is clear by using pŒm ' p! again. ˜ j LEMMA 7.8. Let X n ! X n Rm be the standard inclusion. Given T 2 SD.X n Rm/, the complex j !T is in SD.X /.

PROOF. Looking at the commutative square

... R ...... U2 Œn C m T jU2 ...... ! . ! . . . j . j ...... j !./ ...... !R R ...... ! j U2 Œn C m Š U2\X Œn j T jU2\X ; one checks that (SD1) is fulﬁlled because of the functoriality of j !. Now using that the inverse image functor j is exact, look at Hi.j !T/ Š Hi.j TŒm/ Š Him.j T/ Š j Him.T/: Observe that the last term is zero for i m < .nCm/ or i < n, and so (SD2) holds. Now let i > nN.k/ .n/; k 2. We have

i ! i im H ..j T /jUkC1\X / Š H ..j T Œm/jUkC1\X / Š j H .T jUkC1 /; where the last term is zero due to i m > nN.k/.nCm/ and hence (SD3) holds as well. Finally by [Ban07, Proposition 3.4.5] we have an isomorphism

! ! ! DX j T Œn Š j DX Rm T Œn Š j .DX Rm T Œn C m/ Š j T which is compatible with the orientation, proving (SD4). ˜ Let us now return to the original context. We start with the total space E of a ﬁbre bundle over S n — a topological pseudomanifold of dimension k C n — and a complex T 2 SD.E/. Given a trivializing neighbourhood U S n of the 1 north pole b0 resp. the south pole b1, the restriction of T to .U / Š U F is clearly in SD.U F/ since 1.U / is open. With the preceding lemma we can now deﬁne:

DEFINITION 7.9. Let F ! E ! S n be a ﬁbre bundle as above and T 2 SD.E/. For i D 0; 1, we have inclusions j ...... i...... bi F ... E ...... 1 . . 2 . . i . . i i . . i ......

...... Š ...... U F ..... 1.U / i 272 FILIPP LEVIKOV

2 1 where ji is defined to be the composition ii ı i ı ii . ! 1! ! Set S WD SN WD j0T Š i0 ..0T /jU F / 2 SD.F/, with j0 W b0 F Œ the Š 1 inclusion of the north pole fibre for some trivialization 0 W U F ! .U /. Define SS in the same way using the inclusion of the south pole fibre. In order for S to be well defined over F D bi F with bi 2 U1 \ U2, we have to make an extra assumption, a kind of homogeneity:

DEFINITION 7.10. We call the structure group G of a ﬁbre bundle of the form above adapted to A 2 SD.F/, if for all h 2 G, h!A Š hA Š A.

EXAMPLE 7.11. If F is a Witt space, ICmN .F/ 2 SD.F/. For every stratum- Š preserving automorphism h W F ! F, we have h ICmN .F/ Š ICmN .F/. REMARK 7.12. Assume G to be adapted to S. What if we are given two n 1 trivializing neighbourhoods U1; U2 S with i W Ui F ! .Ui/? We have a commutative diagram 2 ...... i ...... b F ...... U \ U / F F 0 1 2 ...... 2 ...... 1 . 1 ...... h21.b0/ . Š . Š 1 . .U1 \ U2/ E . . 2 ...... 1 ...... i ...... 1 F .. b0 F .. .U1 \ U2/ F where the upper composition is j 1 and the lower composition is j 2. We have to show that .j 1/!T D .j 2/!T. Since G is adapted to S, the transition function 1 ! ! 2 ! 2 ! h21.b0/ preserves S and thereby .j / T D h21.b0/ .j / T Š .j / T . We will need some form of suspension isomorphism for hypercohomology with values in a SD sheaf.

LEMMA 7.13. Let F be a pseudomanifold and S 2 SD.F/. Let p W Rn F ! F be the projection. The inclusion l W 0 F Œ Rn F induces the following isomorphisms on hypercohomology: (a) l W Hk.Rn FI S/ ! HkCn.FI l!S/ k ! k n (b) l! W Hc .FI l S / ! Hc .R FI S / PROOF. Same as for Lemma 6.1, using p ' p!Œn for (a). Note that by b [Bor84, V, 3.13] Rpp A Š A for all A 2 D .X /. ˜ Now we are able to formulate the next proposition and imitate the proof of the Wang sequence given in Section 6.

PROPOSITION 7.14. Let F ! E ! S n be a ﬁbre bundle with a suitable structure group G of automorphisms of the pseudomanifold F, which is adapted to INTERSECTIONHOMOLOGYWANGSEQUENCE 273

SN and SS . Assume E to be canonically stratiﬁed. Given T 2 SD.E/ there are long exact sequences (a) ! Hk.FI S/ ! Hk.EI T/ ! HkCn.FI S/ !, k k kCn (b) ! Hc .FI S / ! Hc .EI T / ! Hc .FI S / !, where S is the self-dual sheaf of Deﬁnition 7.9.

PROOF. Due to Remark 7.12 we need not pay attention to different trivializations. Therefore, in the following proof we will not mention them explicitly. (a) Denote by j the inclusion of the north pole ﬁbre and by i the inclusion of the complement V F. We begin with the distinguished triangle

... ! ...... Rjj T ..T ...... Œ1 ...... Rii T and apply hypercohomology to get ! Hk.FI j !T/ ! Hk.EI T/ ! Hk.V FI i T/ ! : By construction, we have Hk.FI j !T/ Š Hk.FI S/. If i W V F ! E denotes the inclusion, then i T is again isomorphic to a self-dual complex over the product bundle V F containing the south pole ﬁbre b1 F. With the inclusion j1 W b1 F Œ E and using the preceding lemma, we ﬁnally get Hk HkCn ! HkCn .V FI i T / D .FI j1T / Š .FI SS /:

Now choose a trivializing neighbourhood O containing the north pole b0 and the south pole b1. Let l0 and l1 be the corresponding inclusions into O F. We have ! ! ! ! S D j0T D l0.T jOF / Š l1.T jOF / Š j1T D SS ; where the middle isomorphism holds because G is adapted to SN and SS . This completes the proof. (b) Begin with the triangle

. ...... Ri!i T .. T ...... Œ1 ...... Rjj T and apply hypercohomology with compact supports to get

k k k ! Hc .V FIi T / ! Hc .EIT / ! Hc .FIj T / ! : 274 FILIPP LEVIKOV

Now, using part (b) of the preceding lemma, we identify

Hk Hk ! Hk c .V FI i T / Š c .FI j1i T / Š c .FI S /: And again by using j ...... b0 F ... E ...... i1 ...... i2 U F we obtain ! j T Š i1 i2 T Š i1.T jV F /Œn Š S Œn: k kCn Hence the third term in the sequence is equal to Hc .FI j T / Š Hc .FI S /. ˜ The observation S Š SS holds as before. REMARK 7.15. We can still formulate a similar exact sequence even without G being adapted to SN and SS or using the local triviality of the stratifold bundle only. Then, however, the involved SD-complexes over the ﬁbre may be different and depend on the choice of trivializations:

Hk Hk HkCn ! .FI SN / ! .EI T / ! .FI SS / ! ;

Hk Hk HkCn ! c .FI SS / ! c .EI T / ! c .FI SN / ! :

8. Novikov additivity and collapsing of spheres

In [Sie83], Siegel generalizes the classical Novikov additivity of the signature for manifolds to pseudomanifolds satisfying the Witt condition. We want to make a further step forward by dropping the Witt condition in certain cases.7

PROPOSITION 8.1. Let X D X2n X2nk ?, k 2, be a Whitney stratiﬁed compact pseudomanifold with SD.X / ¤ ?. Given subspaces M; E; T X with T a closed neighbourhood of X2nk, such that (1) X D M [ T , (2) M \ T D E, (3) E has a collar in T , and (4) .M; E/ is a compact manifold with boundary. 1 2 Deﬁne X WD M [E c.E/ and X WD T [E c.E/. Then we have the identity .X / D .X 1/ C .X 2/:

7A similar result is given in Theorem 3 of [Hun07]. Thanks to the referee for pointing this out to me. INTERSECTIONHOMOLOGYWANGSEQUENCE 275

PROOF. Once again, consider the formula of Cappell and Shaneson 4.5. Let f W X ! Y be the map collapsing X2nk to a point. The push forward RfT of T 2 SD.E/ is algebraically cobordant to 1 SM˚ Sfcg ICmN .X ; f / ˚ jc ICmN .fcg; f /Œn: SM˚ Here, c is the conepoint f.X2nk/ and jc its inclusion into Y . Since f is 1 constant with rank one, the ﬁrst term is just equal to ICmN .X /. Let us look at Sfcg . f /c. The link L.c/ of c is E, so we deduce from 1 1 2 Ec D f .c L.c// [E c f .L.c// D X and ic 1 c Ec ‹ EcnfcgŠ f .c L.c// Œ X ! (see Section 4B) that S .c/ is a Deligne extension of cS — which is just the 8 restriction of the original self-dual complex. Hence S .c/ 2 SD.X2/. We have Sfcg Hn 2 . f /c D .X I S .c// and consequently Hn Sfcg Hn Sfcg .Y I jc ICmN .fcg; f /Œn/ Š .fcgI ICmN .fcg; f /Œn/ Š Hn.X 2I S.c//: Finally, combining these observations and passing to the signature we get

.X / D .X; Rf.T // 1 1 2 1 2 ˜ D .X ; ICmN .X // C .X ; S .c// D .X / C .X /: Let E be the total space of a fibre bundle over S m as before. We investigate, when the middle hypercohomology group Hn.M; S/ vanishes for a given complex S over a non-Witt M WD cE [E EN of dimension 2n. Since we are only interested in computing the signature, only odd-dimensional spheres are considered here. The strategy is very similar to that of Section 5. PROPOSITION 8.2. Let F 2a ! E ! S 2bC1 be a fibre bundle with F a compact topological pseudomanifold, a 1, b 2. Define M WD cE [E EN , where EN is the total space of the induced fibre bundle

c ./ cF ! EN ! S 2bC1: Given S 2 SD.M /, denote by T the induced element in SD.E/ (compare Section 7). Assume that the following condition (S) is fulﬁlled for the hypercohomology Wang sequence of E:

.aCbC1/ .ab/ H .EI T / “ H .FI SS / is surjective,

8All the axioms are clearly satisﬁed. The only “new” stalk to look at is the one at c, but fcg has even ! codimension and the modiﬁed Deligne extension of c S explained in 4B ensures that SD1-SD4 remain valid. 276 FILIPP LEVIKOV

Then H.aCbC1/.M I S/ D 0: We will need the following vanishing lemma for the hypercohomology of the cone.

LEMMA 8.3. Let X be a 2n-dimensional or a .2n C 1/-dimensional, compact Witt space such that the signature of the pairing over the middle-dimensional intersection homology vanishes. For S 2 SD.cX˚ /, we have

i Hc .cX˚ I S / D 0 for i n C 1:

PROOF. Since S is constructible and cX˚ is a distinguished neighbourhood of the conepoint c, there is (by [Ban07, p. 97], for instance)

i i ! Hc .cX˚ I S / D H .jcS / where the latter is the costalk of S at c. Because of self-duality, however, S satisﬁes the costalk vanishing condition mN .2n C 1/ dim cX˚ C 1 D .n C 1/; H i.j !S/ D 0 for i c mN .2n C 2/ dim cX˚ C 1 D .n C 1/; which is equivalent to the statement. ˜

PROOF OF PROPOSITION 8.2.. Look at the hypercohomology Wang sequence for cF˚ ! EN˚ ! S 2bC1

.aCbC1/ .aCbC1/ N˚ .ab/ ! Hc .cF˚ I U / ! Hc .EI T / ! Hc .cF˚ I U / ! where T D Sj is in SD.EN˚ / and U 2 SD.cF˚ / is constructed as in 7.9. Since EN˚ b 2, we see from the preceding lemma that

.aCbC1/ .aCb/ Hc .cF˚ I U / D Hc .cF˚ I U / D 0 and hence .aCbC1/ N˚ .ab/ Hc .EI T / Š Hc .cF˚ I U /: Decompose M into the open subsets EN˚ and cE˚ with EN˚ \ cE˚ D E .0; 1/. Consider the Mayer–Vietoris hypercohomology sequence (see [Ive86, III.7.5] or [Bre97, II, ~ 13], for instance)

i .aCbC1/ aCbC1 Hc .E .0; 1/I S j/ ! .aCbC1/ .aCbC1/ N˚ Hc .cE˚ I S j/ ˚ Hc .EI S j/ i .aCbC1/ .aCb/ aCb Hc .M I S / Hc .E .0; 1/I S j/ ! INTERSECTIONHOMOLOGYWANGSEQUENCE 277

Let us ﬁrst show that iaCb is injective. We use the following exact sequence for hypercohomology with compact supports (see [Ive86, III.7.7])

.aCbC1/ .aCb/ .aCb/ !Hc .fcgI S /!Hc .cE˚ nfcgI S /!Hc .cE˚ I S /! : It sufﬁces to show that

.aCbC1/ .aCbC1/ Hc .fcgI S / Š H .fcgI S /

.aCbC1/ vanishes. The latter is isomorphic to H .Sc/ which is 0 because of .aCbC1/ (SD3). Due to the preceding lemma Hc .cE˚ I S j/ is 0. The surjectivity of iaCbC1 follows now from the condition (S) using the naturality of the hypercohomology Wang sequence in completely the same manner as in the proof of Proposition 5.2. ˜

REMARK 8.4. As you can see, we have only used the vanishing lemma 8.3, which is valid for every S 2 SD.F/. Hence, in the proposition, the structure group G need not be adapted.

REMARK 8.5. In the case of a Witt ﬁbre F the condition (S) is the one of Section 5. See the examples there, especially 5.6.

COROLLARY 8.6. Let M be as in 8.2. The signature .M / of M vanishes.

COROLLARY 8.7. Let X be as in 8.1 such that Xnk is an odd-dimensional sphere. Assume E to satisfy (S). Then .X / D .X 1/ D .M;@M /, where the latter is the Novikov signature of M .

Acknowledgements

The article is based on the author’s Diploma thesis at the University of Hei- delberg [Lev07] and is motivated by the MSRI workshop on the Topology of Stratiﬁed Spaces. I want to express my gratitude to Greg Friedman, Eugenie´ Hunsicker, Anatoly Libgober and Laurentiu-George Maxim for the organisation of this event, the accompanying support and the opportunity for giving a talk. I am also indebted to my thesis advisor Markus Banagl for his help and advice during the development of the thesis. Last but not least I am grateful to the referee for the careful reading of the preliminary version.

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FILIPP LEVIKOV INSTITUTE OF MATHEMATICS KING’S COLLEGE ABERDEEN AB24 3UE SCOTLAND [email protected]