The Equivariant L-Class of Pseudomanifolds

TL;DR

This paper develops an equivariant L-class for stratified pseudomanifolds with group actions, extending classical invariants to singular spaces and providing formulas for various group action scenarios.

Contribution

It introduces an equivariant L-class for pseudomanifolds with group actions, generalizing the Goresky-MacPherson L-class and establishing new product and quotient formulas.

Findings

Defines equivariant L-class in rational homology

Provides a product formula under the equivariant K"unneth isomorphism

Shows the class coincides with classical L-class for free actions

Abstract

We construct an equivariant L-class for orientation preserving actions of a compact Lie group on a Whitney stratified compact oriented pseudomanifold that satisfies the Witt condition, for example on a compact pure-dimensional complex algebraic variety. The class lies in equivariant rational homology and its restriction to the trivial group is the Goresky-MacPherson L-class. For a smooth action on a manifold, the class is equivariantly Poincar\'e dual to the Hirzebruch L-class of the Borel homotopy quotient of the tangent bundle. We also provide a product formula under the equivariant K\"unneth isomorphism. If the group acts freely, then the equivariant L-class identifies with the Goresky-MacPherson L-class of the orbit space. The construction method rests on establishing Whitney (B)-regularity for finite-dimensional compact pseudomanifold approximations to the Borel construction, and on the author's Verdier-Riemann-Roch type formulae for the Goresky-MacPherson L-class.