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Additivity and non-additivity for perverse signatures

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posted on 2015-04-01, 11:48 authored by Greg Friedman, Eugenie Hunsicker
A well-known property of the signature of closed oriented 4n-dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifolds with boundary. Wall showed that this property is not true of signatures on manifolds with boundary and that the difference from additivity could be described as a certain Maslov triple index. Perverse signatures are signatures defined for any oriented stratified pseudomanifold, using the intersection homology groups of Goresky and MacPherson. In the case of Witt spaces, the middle perverse signature is the same as the Witt signature. This paper proves a generalization to perverse signatures of Wall's non-additivity theorem for signatures of manifolds with boundary. Under certain topological conditions on the dividing hypersurface, Novikov additivity for perverse signatures may be deduced as a corollary. In particular, Siegel's version of Novikov additivity for Witt signatures is a special case of this corollary.

Funding

The authors are partially supported by MSRI.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK

Volume

676

Pages

51 - 95 (45)

Citation

FRIEDMAN, G. and HUNSICKER, E., 2013. Additivity and non-additivity for perverse signatures. Journal für die reine und angewandte Mathematik, 676, pp. 51 - 95.

Publisher

© Walter de Gruyter

Version

  • VoR (Version of Record)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2013

Notes

This article was published in the Journal für die reine und angewandte Mathematik [© Walter de Gruyter]. It is also available at: http://dx.doi.org/10.1515/crelle.2012.005

ISSN

0075-4102

Language

  • en

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