Signature homology

Minatta, Augusto

doi:10.1515/crelle.2006.024

Cited by 3 publications

(4 citation statements)

References 6 publications

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“…We will define a bordism between stratified pseudomanifolds with self-dual mezzoperversities, that is Cheeger spaces. The corresponding topological object was introduced by Banagl in [Ban02] where it was denoted Ω SD * , and was later considered by Minatta [Min06] where it was denoted Sig, short for 'signature homology.' Definition 4.7.…”

Section: Bordism Invariance Of the Analytic Signaturementioning

confidence: 99%

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The Novikov conjecture on Cheeger spaces

Albin¹,

Leichtnam²,

Mazzeo³

et al. 2017

J. Noncommut. Geom.

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We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an L 2 -de Rham cohomology theory satisfying Poincaré duality. We prove that this cohomology theory is invariant under stratified homotopy equivalences and that its signature is invariant under Cheeger space cobordism. Analogous results, after coupling with a Mischenko bundle associated to any Galois covering, allow us to carry out the analytic approach to the Novikov conjecture: we define higher analytic signatures of a Cheeger space and prove that they are stratified homotopy invariants whenever the assembly map is rationally injective. Finally we show that the analytic signature of a Cheeger space coincides with its topological signature as defined by Banagl. ContentsThe authors are happy to thank Markus Banagl, Jochen Brüning, Jeff Cheeger, Greg Friedman, Michel Hilsum and Richard Melrose for many useful and interesting discussions. Thanks are also due to the referee for a careful reading of the original manuscript and for interesting remarks. 1.Smoothly stratified spaces, iie metrics, and mezzoperversities 1.1. Smoothly stratified spaces. There are many notions of stratified space [Klo09]. Most of them agree that a stratified space X is a locally compact, second countable, Hausdorff topological space endowed with a locally finite (disjoint) decompositionwhose elements, called strata, are locally closed and verify the frontier condition:The depth of a stratum Y is the largest integer k such that there is a chain of strata Y = Y k , . . . , Y 0 with Y j ⊂ Y j−1 for 1 ≤ j ≤ k. A stratum of maximal depth is always a closed manifold. The maximal depth of any stratum in X is called the depth of X as a stratified space. Thus a stratified space of depth 0 is a smooth manifold with no singularity.Where the definitions differ is on how much regularity to impose on the strata and how the strata 'fit together'. In this paper we shall mainly consider smoothly stratified pseudomanifolds with Thom-Mather control data. We proceed to recall the definition, directly taken from [BHS91] and [ALMP12].Definition 1.1. A smoothly stratified space of depth 0 is a closed manifold. Let k ∈ N, assume that the concept of smoothly stratified space of depth ≤ k has been defined. A smoothly stratified space X of depth k + 1 is a locally compact, second countable Hausdorff space which admits a locally finite decomposition into a union of locally closed strata S = {Y j }, where each Y j is a smooth (usually open) manifold, with dimension depending on the index j. We assume the following:

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Section: Bordism Invariance Of the Analytic Signaturementioning

confidence: 99%

“…Minatta [Min06,Theorem 3.4] has shown that M → Sig top * (M ) is a multiplicative generalized homology theory (with the product induced by Cartesian product) and by adding a formal variable t, the signature becomes an isomorphism of graded rings…”

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confidence: 99%

The Novikov conjecture on Cheeger spaces

Albin¹,

Leichtnam²,

Mazzeo³

et al. 2017

J. Noncommut. Geom.

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“…One 15 of the defining properties of these spaces is precisely the vanishing of the signatures of links; these signatures are defined with respect to the sheaf cohomology of the self-dual sheaves (restricted to the link). Bordism of L-spaces is studied in [4], and the associated bordism homology theory, dubbed "signature homology," was introduced by Minatta in [26]; see also [5]. For more on the general philosophy of constructing bordism theories of this type, see Banagl's survey article [7].…”

Section: Examplesmentioning

confidence: 99%

Stratified and unstratified bordism of pseudomanifolds

Friedman¹

2015

Topology and its Applications

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We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples of pseudomanifold bordism theories, such as bordism of Witt spaces and IP spaces. Along the way, we relate the stratified and unstratified points of view for describing various classes of pseudomanifolds.

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“…Our chief tool is piecewise-linear signature homology S PL * (−) as introduced in [5]. Based on M. Kreck's topological stratifolds as geometric cycles, and using topological transversality, topological signature homology was first introduced by A. Minatta in [31]. The coefficients had already been introduced earlier in [3,Chapter 4].…”

Section: Introductionmentioning

confidence: 99%

Topological and Hodge L-classes of singular covering spaces and varieties with trivial canonical class

Banagl

2018

Geom Dedicata

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The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension (so-called L-pseudomanifolds). This class of spaces contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We apply this result in proving the Brasselet-Schürmann-Yokura conjecture for normal complex projective 3-folds with at most canonical singularities, trivial canonical class and positive irregularity. The conjecture asserts the equality of topological and Hodge L-class for compact complex algebraic rational homology manifolds.

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Signature homology

Cited by 3 publications

References 6 publications

The Novikov conjecture on Cheeger spaces

The Novikov conjecture on Cheeger spaces

Stratified and unstratified bordism of pseudomanifolds

Topological and Hodge L-classes of singular covering spaces and varieties with trivial canonical class

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