Description of the vector $G$-bundles over $G$-spaces with quasi-free proper action of discrete group $G$

We give a description of the vector $G$-bundles over $G$-spaces with quasi-free proper action of discrete group $G$ in terms of the classifying space.

💡 Research Summary

The paper addresses the classification problem for vector bundles equipped with an action of a discrete group (G) on a base space (M) under the hypothesis that the action is “quasi‑free”: there exists a normal finite subgroup (H\lhd G) (the stationary subgroup) such that the induced action of the quotient group (G_{0}=G/H) on (M) is free, while (H) may have fixed points on the total space of the bundle.

First, the authors recall the classical Conner–Floyd construction, which reduces the study of equivariant bordisms to the analysis of fixed‑point sets and lower‑rank bordisms. In this setting, a (G)‑equivariant vector bundle (\xi\to M) can be decomposed using the complete set of irreducible unitary representations (\rho_{k}:H\to U(V_{k})). Explicitly, \