Classification of equivariant vector bundles over two-sphere

TL;DR

This paper provides a comprehensive classification of topological equivariant complex vector bundles over the two-sphere under compact Lie group actions, using equivariant Chern classes and isotropy representations, with calculations of equivariant homotopy of clutching maps.

Contribution

It offers the first exhaustive classification of equivariant vector bundles over the two-sphere, including calculations of equivariant homotopy and conditions for classification.

Findings

Inequivariant Chern classes and isotropy representations suffice for classification.

Explicit calculation of equivariant homotopy of clutching maps.

Classification extends to other surfaces like the two-torus and real projective plane.

Abstract

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