Hodge theory for intersection space cohomology

Banagl, Markus; Hunsicker, Eugénie

doi:10.2140/gt.2019.23.2165

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Hodge theory for intersection space cohomology

Markus Banagl and Eugénie Hunsicker

Geometry & Topology 23 (2019) 2165–2225

DOI: 10.2140/gt.2019.23.2165

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 $L^{2}$ 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/

Volume 23, issue 5 (2019)