Twistors, Self-Duality, and Spin$^c$ Structures

TL;DR

This paper presents a new, simpler geometric proof that every compact oriented 4-manifold admits spin$^c$ structures using twistor spaces, and explores their generalization to higher dimensions.

Contribution

It introduces a novel, geometric proof for the existence of spin$^c$ structures on 4-manifolds and extends these ideas to higher-dimensional cases.

Findings

New geometric proof of spin$^c$ structures on 4-manifolds

Clarification of four-dimensional geometry aspects

Extension of spin$^c$ concepts to higher dimensions

Abstract

The fact that every compact oriented 4-manifold admits spin$^c$ structures was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither direct nor transparent. This article gives a new proof using twistor spaces that is simpler and more geometric. After using these ideas to clarify various aspects of four-dimensional geometry, we then explain how related ideas can be used to understand both spin and spin$^c$ structures in any dimension.