Bundle transfer of L–homology orientation classes for singular spaces

Banagl, Markus

doi:10.2140/agt.2024.24.2579

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Bundle transfer of $L$–homology orientation classes for singular spaces

Markus Banagl

Algebraic & Geometric Topology 24 (2024) 2579–2618

DOI: 10.2140/agt.2024.24.2579

Abstract

We consider transfer maps on ordinary homology, bordism of singular spaces and homology with coefficients in Ranicki’s symmetric $L$ –spectrum, associated to block bundles with closed oriented PL manifold fiber and compact polyhedral base. We prove that if the base polyhedron is a Witt space, for example a pure-dimensional compact complex algebraic variety, then the symmetric $L$ –homology orientation of the base, constructed by Laures, McClure and the author, transfers to the $L$ –homology orientation of the total space. We deduce from this that the Cheeger–Goresky–MacPherson $L$ –class of the base transfers to the product of the $L$ –class of the total space with the cohomological $L$ –class of the stable vertical normal microbundle.

Keywords

transfer homomorphisms, stratified spaces, intersection homology, characteristic classes, algebraic $L$–Theory, block bundles, cobordism

Mathematical Subject Classification

Primary: 55N33, 55R12, 57N80, 57Q50, 57R20

Secondary: 57Q20

References

Publication

Received: 8 April 2022

Revised: 9 August 2023

Accepted: 30 August 2023

Published: 19 August 2024

Authors

Markus Banagl
Institut für Mathematik Ruprecht-Karls-Universität Heidelberg Heidelberg Germany

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Volume 24, issue 5 (2024)