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

Michael R. Klug; Luuk Stehouwer

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arXiv · 2112.07290

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

Michael R. Klug·Luuk Stehouwer

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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.

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Michael R. Klug, Luuk Stehouwer. 2021-12-14. Some properties of $\operatorname{Pin}^\pm$-structures on compact surfaces. https://arxiv.org/abs/2112.07290

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