Bundle Transfer of L-Homology Orientation Classes for Singular Spaces

TL;DR

This paper proves that L-homology orientations transfer from base to total space in block bundles over Witt spaces, including complex algebraic varieties, linking L-classes of base and total space.

Contribution

It establishes the transfer of L-homology orientations and classes in singular spaces with Witt space bases, extending previous theories to new classes of spaces.

Findings

L-homology orientation transfers to total space in block bundles over Witt spaces.

L-classes of base and total space are related via transfer, including cohomological classes.

The results apply to complex algebraic varieties as examples of Witt spaces.

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.