Bordisms of manifolds with proper action of a discrete group: signatures and descriptions of $G$-bundles

TL;DR

This paper introduces an equivariant signature invariant for manifolds with proper discrete group actions, reducing its computation to fixed point sets, and describes equivariant vector bundles in this context using classifying spaces.

Contribution

It defines the equivariant signature as a bordism invariant and provides a novel description of equivariant vector bundles under proper quasi-free actions using classifying spaces.

Findings

Equivariant signature is an invariant of equivariant bordisms.

Computation reduces to fixed point sets with tubular neighborhoods.

Description of equivariant vector bundles via classifying spaces.

Abstract

In this work the equivariant signature of a manifold with proper action of a discrete group is defined as an invariant of equivariant bordisms. It is shown that the computation of this signature can be reduced to its computation on fixed points sets equipped with their tubular neighborhoods. It is given a description of the equivariant vector bundles with action of a discrete group $G$ for the case when the action over the base is proper quasi-free, i.e. the stationary subgroup of any point is finite. The description is given in terms of some classifying space.