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In his paper "Spin structures and quadratic forms on surfaces", Johnson constructs a bijection between the set of spin strucutres on a smooth closed orientable surface $S$ and the set of quadratic forms on $H_1(S,\mathbb{Z}_2)$.

It seems to me that this result (and all the proofs) extends to the case of a not closed orientable surface. Is it true ?

flag	asked Sep 22 2011 at 12:03 kieffer 83●2

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If the surface is not closed, then $H_1(S)$ is no longer a symplectic vector space (over $\mathbb F_2$), so I'm not sure what it would mean to be a quadratic form. – Sam Gunningham Sep 22 2011 at 13:13

When I say quadratic forms in the case of not closed surfaces I'm looking to the set of mappings $H_1(S,\mathbb{Z}_2)\rightarrow \mathbb{Z}_2$ such that $\omega(a+b)=\omega(a)+\omega(b) +(a \cdot b)$ where $(a \cdot b)$ is the intersection of $a$ and $b$ in $H_1(S,\mathbb{Z}_2)$. – kieffer Sep 22 2011 at 15:33

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Yes, you get a quadratic form via Johnson's construction in this way, but it is singular (the radical of the bilinear form is given by $\mathrm{Im}(H_1(\partial S;\mathbb{Z}/2) \to H_1(S;\mathbb{Z}/2))$). If you insist that the Spin structure along the boundary is fixed, I explained below that this doesn't quite work. If you don't insist that it is fixed, you obtain a bijection between spin structures on S and quadratic refinements of this singular bilinear form. – Oscar Randal-Williams Sep 22 2011 at 17:46

Oscar Randal-Williams · Accepted Answer · 2011-09-22 13:33:10Z UTC

It is not quite true. There is an algebraic gadget one can produce from a Spin structure on a surface with boundary (where we fix the Spin structure along the boundary), and one gets a bijection from Spin structures to such gadgets, but the process is horribly non-canonical. It does however let you prove that there are precisely two isomorphism classes of Spin surfaces of each topological type (for genus $\geq 2$).

This is discussed in Section 2.3 of my paper "Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces" (http://arxiv.org/abs/1001.5366), albeit it more generality (so-called $r$-Spin structures).

Briefly, the point is this: one ought to have some sort of algebraic structure on $H_1(S, \partial S ;\mathbb{Z}/2)$, and Johnson's construction works fine for elements represented by unions of simple closed curves, but for arcs between boundary components there is all sorts of indeterminacy and choices to be made.

Spin structures and quadratic forms on surfaces

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