Topological Bordism of Singular Spaces and an Application to Stratified L-Classes

TL;DR

This paper develops a new generalized homology bordism theory for stratified spaces, enabling the definition of topologically invariant L-classes that extend classical invariants to broader classes of singular spaces.

Contribution

It introduces a flexible geometric framework for bordism of stratified spaces and constructs topologically invariant L-classes applicable to Witt-spaces with singularities.

Findings

Constructed a generalized homology bordism theory for stratified spaces.

Defined topologically invariant L-classes on certain singular spaces.

Showed these L-classes coincide with classical Goresky-MacPherson classes on pseudomanifolds.

Abstract

A generalized-homology bordism-theory is constructed, such that for certain manifold homotopy stratified sets (MHSS; Quinn-spaces) homeomorphism-invariant geometric fundamental-classes exist. The construction combines three ideas: Firstly, instead of restricting geometric cycles by conditions on links only, a more flexible framework is built directly via geometric properties, secondly, controlled topology methods are used to give an accessible link-based criterion to detect suitable cycles and thirdly, a geometric argument is used to show, that these classes of cycles are suitable to study the transition to intrinsic stratifications. As an application, we give a construction of topologically (homeomorphism) invariant (homological) L-classes on MHSS Witt-spaces satisfying conditions on Whitehead-groups of links and the dimensional spacing of meeting strata. These L-classes agree, whenever those spaces are additionally pl-pseudomanifolds, with the Goresky-MacPherson L-classes.