Some properties of $\operatorname{Pin}^\pm$-structures on compact surfaces

TL;DR

This paper investigates the relationship between $ ext{Pin}^ ext{±}$-structures and diffeomorphisms on compact surfaces, establishing cobordism as the key equivalence and exploring the count of structures within cobordism classes.

Contribution

It proves that two $ ext{Pin}$-structures on a surface differ by a diffeomorphism if and only if they are cobordant, extending known results from $ ext{Spin}$-structures and showing limitations in higher dimensions.

Findings

Two $ ext{Pin}$-structures differ by a surface diffeomorphism iff they are cobordant.

The cobordism relation does not extend to higher dimensions.

The number of $ ext{Pin}$-structures in a given cobordism class is explicitly counted.

Abstract

We show that two $\operatorname{Pin}$-structures on a surface differ by a diffeomorphism of the surface if and only if they are cobordant (for comparison, the analogous fact has already been shown for $\operatorname{Spin}$-structures). We give a construction that shows that this does not extend to dimensions greater than two. In addition, we count the number of $\operatorname{Pin}$-structures on a surface in a given cobordism class.