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Topological bordism of singular spaces and an application to stratified L-classes

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 lin...

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Bibliographic Details
Main Author: Rabel, Martin (Author)
Format: Article (Journal)
Language:English
Published: 1 January 2024
In: Topology and its applications
Year: 2024, Volume: 341, Pages: 1-26
DOI:10.1016/j.topol.2023.108744
Online Access:Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1016/j.topol.2023.108744
Verlag, lizenzpflichtig, Volltext: https://www.sciencedirect.com/science/article/pii/S0166864123003371
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Author Notes:Martin Rabel

MARC

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520 |a 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. 
650 4 |a Bordism 
650 4 |a Controlled topology 
650 4 |a Generalized homology-theories 
650 4 |a Homotopically stratified spaces 
650 4 |a L-classes 
650 4 |a Singular spaces 
650 4 |a Stratified transversality 
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