Intersection Spaces, Equivariant Moore Approximation and the Signature

Markus Banagl and Bryce Chriestenson

Journal of Singularities
volume 16 (2017), 141-179

Received 9 May 2017.

DOI: 10.5427/jsing.2017.16g

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Abstract:

We generalize the first author's construction of intersection spaces to the case of stratified pseudomanifolds of stratification depth 1 with twisted link bundles, assuming that each link possesses an equivariant Moore approximation for a suitable choice of structure group. As a by-product, we find new characteristic classes for fiber bundles admitting such approximations. For trivial bundles and flat bundles whose base has finite fundamental group these classes vanish. For oriented closed pseudomanifolds, we prove that the reduced rational cohomology of the intersection spaces satisfies global Poincar\'e duality across complementary perversities if the characteristic classes vanish. The signature of the intersection spaces agrees with the Novikov signature of the top stratum. As an application, these methods yield new results about the Goresky-MacPherson intersection homology signature of pseudomanifolds. We discuss several nontrivial examples, such as the case of flat bundles and symplectic toric manifolds.

Keywords:

Stratified spaces, pseudomanifolds, intersection homology, Poincaré duality, signature, fiber bundles

2010 Mathematical Subject Classification:

55N33, 57P10, 55R10, 55R70

Author(s) information:

Markus Banagl	Bryce Chriestenson
Mathematisches Institut	Department of Mathematics
Ruprecht-Karls-Universität Heidelberg	Western Oregon University
Im Neuenheimer Feld 205	Monmouth OR 97361, USA
69120 Heidelberg, Germany
email: banagl@mathi.uni-heidelberg.de	email: chriestensonb@mail.wou.edu