Twistors, Self-Duality, and Spinᶜ Structures

Abstract

The fact that every compact oriented 4-manifold admits spinᶜ 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 spinᶜ structures in any dimension.