Classification of equivariant vector bundles over two-torus

TL;DR

This paper provides a comprehensive classification of topological equivariant complex vector bundles over the two-torus under group actions, using Chern classes and isotropy representations, with implications for other surfaces.

Contribution

It offers the first exhaustive classification of equivariant vector bundles over the two-torus, detailing the role of Chern classes and isotropy at key points.

Findings

Inequivariant Chern classes and isotropy representations suffice for classification.

Homotopy of equivariant clutching maps is computed.

Classification extends to real projective plane and Klein bottle.

Abstract

We exhaustively classify topological equivariant complex vector bundles over two-torus under a compact Lie group (not necessarily effective) action. It is shown that inequivariant Chern classes and isotropy representations at (at most) six points are sufficient to classify equivariant vector bundles except a few cases. To do it, we calculate homotopy of the set of equivariant clutching maps. And, classification on real projective plane, Klein bottle will appear soon