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 complex vector bundles equipped with an action of a discrete group G on a G‑space X, under the hypothesis that the action is proper and quasi‑free (i.e., each stabilizer Gₓ is finite and the orbit space X/G is Hausdorff). The author’s main achievement is to express the equivariant isomorphism classes of such G‑bundles in terms of ordinary homotopy classes of maps from the classifying space BG to the classifying space BU(n) of rank‑n complex vector bundles, but with a crucial refinement that records the local representation data of the stabilizer groups.

The exposition begins by fixing the topological framework. A proper quasi‑free action guarantees that the quotient X/G inherits a reasonable CW‑structure and that the stabilizers are finite. This allows the construction of the universal free G‑space EG (a contractible CW‑complex with a free G‑action) and its quotient BG=EG/G, the classifying space of G. The key observation is that the Borel construction EG×_G X provides a G‑equivariant homotopy equivalence between the original G‑space X and a fiber bundle over BG with fiber X. Consequently, any G‑equivariant vector bundle E→X can be pulled back to a G‑invariant bundle over EG×_G X, which in turn descends to an ordinary vector bundle over BG.

The next step is to translate the equivariant data into a map f: BG→BU(n). For a free G‑action the correspondence is classical: equivariant bundles correspond bijectively to homotopy classes