A Fundamental Class for Intersection Spaces of Depth One Witt Spaces

Abstract

By a theorem of Banagl-Chriestenson, intersection spaces of depth one pseudomanifolds exhibit generalized Poincaré 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 Poincaré 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 Poincaré 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.

A short note on simplicial stratifications

“…The method of intersection spaces assigns generalized reduced Poincaré duality spaces depending on a perversity to certain stratified pseudomanifolds by performing a homotopy theoretic modification in a neighborhood of the singular set. The missing fundamental class of these generalized reduced Poincaré duality spaces has been constructed for certain depth one stratifications (see [17] and [20]). By construction, intersection spaces of stratified spaces depend on the choice of homology truncations (Moore approximations) of (recursively modified) links.…”