A fundamental class for intersection spaces of depth one Witt spaces

TL;DR

This paper demonstrates that the middle-perversity intersection space of a depth one Witt space can be turned into a rational Poincaré duality space through a single cell attachment, under certain homological conditions.

Contribution

It extends previous work by showing how to achieve rational Poincaré duality for intersection spaces of depth one Witt spaces using cell attachments and rational homology conditions.

Findings

The rational Hurewicz homomorphism condition implies vanishing characteristic classes.

The converse holds under certain dimension bounds.

The signature comparison relates the constructed space to intersection homology.

Abstract

By a theorem of Banagl-Chriestenson, intersection spaces of depth one pseudomanifolds exhibit generalized Poincar\'{e} duality of Betti numbers, provided that certain characteristic classes of the link bundles vanish. In this paper, we show that the middle-perversity intersection space of a depth one Witt space can be completed to a rational Poincar\'{e} duality space by means of a single cell attachment, provided that a certain rational Hurewicz homomorphism associated to the link bundles is surjective. Our approach continues previous work of Klimczak covering the case of isolated singularities with simply connected links. For every singular stratum, we show that our condition on the rational Hurewicz homomorphism implies that the Banagl-Chriestenson characteristic classes of the link bundle vanish. Moreover, using Sullivan minimal models, we show that the converse implication holds at least in the case that twice the dimension of the singular stratum is bounded by the dimension of the link. As an application, we compare the signature of our rational Poincar\'{e} duality space to the Goresky-MacPherson intersection homology signature of the given Witt space. We discuss our results for a class of Witt spaces having circles as their singular strata.