Spin structures on complex projective spaces and circle actions

TL;DR

This paper offers an alternative proof that even-dimensional complex projective spaces do not admit spin structures, utilizing circle actions to establish the result.

Contribution

It provides a new proof technique for the non-existence of spin structures on even-dimensional complex projective spaces using circle actions.

Findings

Complex projective spaces $ ext{CP}^{2m}$ do not admit spin structures.

Circle actions can be used to prove topological properties of manifolds.

The proof offers an alternative to existing methods for understanding spin structures.

Abstract

It is known that the complex projective space $\mathbb{CP}^n$ admits a spin structure if and only if $n$ is odd. In this paper, we provide another proof that $\mathbb{CP}^{2m}$ does not admit a spin structure, by using a circle action.