Classification of equivariant vector bundles over real projective plane

TL;DR

This paper classifies equivariant topological complex vector bundles over the real projective plane under group actions, extending known classifications from the two-sphere via covering maps, with a focus on Chern classes and isotropy representations.

Contribution

It provides a comprehensive classification of equivariant vector bundles over the real projective plane, linking it to the classification over the two-sphere and identifying key invariants.

Findings

Classification is complete except for one case.

Chern classes and isotropy representations at up to three points suffice.

The problem reduces to the classification over the two-sphere.

Abstract

We classify equivariant topological complex vector bundles over real projective plane under a compact Lie group (not necessarily effective) action. It is shown that nonequivariant Chern classes and isotropy representations at (at most) three points are sufficient to classify equivariant vector bundles over real projective plane except one case. To do it, we relate the problem to classification on two-sphere through the covering map because equivariant vector bundles over two-sphere have been already classified.