3
$\begingroup$

It is well known that the spin structures on an oriented surface (with boundary) $M$ are in bijection with the set of cohomology classes $H^1(M,\mathbb{Z}/2)$. By Lefschetz duality, these correspond to homology classes $H_1(M, \partial M; \mathbb{Z}/2)$. This means that one could represent spin structures on a surface by drawing embedded curves.

Are there any references discussing this? How could one compute the boundary restrictions of a spin structure on a surface $M$ starting with a set of 1-dimensional submanifolds of $M$ representing a particular homology class?

Improve this question
$\endgroup$
4
  • 3
    $\begingroup$ In my experience, it is not common to think of spin structures like this. The issue is that to establish the bijection from your first paragraph, you have to first fix a trivialization of the tangent bundle. This is unnatural, and for instance means the action of the mapping class group on the set of spin structures is not natural with respect to this parameterization. I think the best geometric way to parameterize spin structures on surfaces is via quadratic forms, as in Dennis Johnson’s paper on spin structures (which I recommend reading if you haven’t already). $\endgroup$
    – Andy Putman
    Commented Feb 23, 2023 at 2:48
  • 2
    $\begingroup$ I share Andy's view, reading 'D. Johnson, Spin structures and quadratic forms on surfaces, J. LMS (2), 22 (1980), 365–373' is a must for a geometric understanding of spin structures on surfaces. $\endgroup$
    – Sebastian
    Commented Feb 23, 2023 at 3:22
  • $\begingroup$ I vaguely recall spin structures being introduced this way in the textbook Superstring theory (Green, Schwartz, Witten) but lack a copy right now $\endgroup$
    – AlexArvanitakis
    Commented Feb 23, 2023 at 3:39
  • $\begingroup$ Set of spin structures is just a torsor over first mod 2 cohomology. If you work a bit (I did a few years ago but totally forgot those constructions), you can cook up an example of a) diffeo that induces Id on mod 2 cohomology, but act nontrivially on the set of spin structures, and vice versa b) spin diffeo that acts on mod 2 cohomology nontrivially. In my opinion, those facts make choice of any particular bijection between those two sets pretty useless. $\endgroup$
    – Denis T
    Commented Feb 23, 2023 at 6:37

0

Reset to default

You must log in to answer this question.