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

Abstract

We show that two $\mathrm{Pin}$-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 $\mathrm{Spin}$-structures). We give a construction that shows that this does not extend to dimensions greater than two. In addition, we count the number of $\mathrm{Pin}$-structures on a surface in a given cobordism class.

10 pages, 1 figure