Volume 23, issue 5 (2019)

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Hodge theory for intersection space cohomology

Markus Banagl and Eugénie Hunsicker

Geometry & Topology 23 (2019) 2165–2225
Abstract

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincaré duality across complementary perversities. The resulting homology theory is well known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted L2 harmonic forms on the regular part, equipped with a fibred scattering metric. Some consequences of our methods for the signature are discussed as well.

Keywords
conifold transition, fibred cusp metrics, harmonic forms, Hodge theorem, intersection cohomology, intersection form, intersection space, $L^2$ cohomology, perversity function, Poincaré duality, pseudomanifolds, scattering metrics, signature theorem, stratified space
Mathematical Subject Classification 2010
Primary: 55N33, 58A14
References
Publication
Received: 20 February 2015
Revised: 4 August 2017
Accepted: 11 October 2018
Published: 13 October 2019
Proposed: Steve Ferry
Seconded: Frances Kirwan, Ralph Cohen
Authors
Markus Banagl
Mathematisches Institut
Universität Heidelberg
Heidelberg
Germany
http://www.mathi.uni-heidelberg.de/~banagl/
Eugénie Hunsicker
Department of Mathematical Sciences
Loughborough University
Loughborough
United Kingdom
https://www.lboro.ac.uk/departments/maths/staff/academic/eug\% c3\% a9nie-hunsicker/