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

Abstract

We show that two 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 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 Pin-structures on a surface in a given cobordism class.