# Reference request: Spin structures on surfaces and the spin mapping class group

I am looking for references on the following: Spin structures on surfaces, and particularly the spin mapping class group.

What is known about generating the spin mapping class group? Has anybody found a finite set of generators?

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A warning: there are (at least) two things that people call the Spin mapping class group: one is what Andy has described below, and another is a certain central $\mathbb{Z}/2$ extension of it. From the point of view of finite generation of course it makes little difference, but it is worth being aware of when citing results. – Oscar Randal-Williams Jul 11 '13 at 17:01

The answer of course depends on the spin structure chosen. The paper

Johnson, Dennis, Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2) 22 (1980), no. 2, 365–373.

proves that the set of spin structures on a closed surface $\Sigma_g$ of genus $g$ can be identified with the set of quadratic forms on $H_1(\Sigma_g;\mathbb{Z}/2)$. Fix a symplectic basis $a_1,b_1,\ldots,a_g,b_g$ for $H_1(\Sigma_g;\mathbb{Z}/2)$. The arf invariant of a quadratic form $q$ on $H_1(\Sigma_g;\mathbb{Z}/2)$ is

$$\text{Arf}(q) = \sum_{i=1}^g q(a_i) q(b_i) \in \mathbb{Z}/2.$$

It does not depend on the choice of symplectic basis. Up to isomorphism, these quadratic forms are classified by their Arf invariant.

The spin mapping class group associated to a quadratic form $q$ as above is the stabilizer $\text{Mod}_g(q)$ of $q$. Let $\text{Mod}_g[2]$ be the level $2$ subgroup of the mapping class group, that is, the kernel of the action of $\text{Mod}_g$ on $H_1(\Sigma_g;\mathbb{Z}/2)$. Also, let $\Gamma(q) < Sp(2g,\mathbb{Z}/2)$ be the stabilizer of $q$. We have a short exact sequence

$$1 \longrightarrow \text{Mod}_g[2] \longrightarrow \text{Mod}_g(q) \longrightarrow \Gamma(q) \longrightarrow 1.$$

To determine generating sets of $\text{Mod}_g(q)$, therefore, we need generating sets for $\text{Mod}_g[2]$ and $\Gamma(q)$. Proposition 2.1 of

Humphries, Stephen P., Normal closures of powers of Dehn twists in mapping class groups. Glasgow Math. J. 34 (1992), no. 3, 313–317.

says that $\text{Mod}_g[2]$ is generated by squares of Dehn twists (though it is finitely generated, I am not aware of an explicit finite set of squares of Dehn twists that generate it). As for $\Gamma(q)$, I am only aware of an explicit generating set for it when $q$ is the standard quadratic form

$$q(\sum_{i=1}^g (c_i a_i + d_i b_i)) = \sum_{i=1}^g c_i d_i.$$

In this case, Proposition 14 of

Dieudonne, J.: Sur les Groupes Classiques. Publications de l'Institut de Mathematique de l'Univer-sity de Strasbourg VI, Hermann, Paris, 1967.

says that $\Gamma(q)$ is generated by the set of anisotropic transvections, that is, transvections about elements $v \in H_1(\Sigma_g;\mathbb{Z}/2)$ such that $q(v) \neq 0$. It shouldn't be hard to work out generators for the other isomorphism class of quadratic forms from this.

There are, of course, many other things known about the spin mapping class group. As far as its simple combinatorial group theory goes, I'll just mention one result, namely that its abelianization is $\mathbb{Z}/4$. It is proven in

Harer, John L., The rational Picard group of the moduli space of Riemann surfaces with spin structure. Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), 107–136, Contemp. Math., 150, Amer. Math. Soc., Providence, RI, 1993.

that its abelianization is at most $\mathbb{Z}/4$. He refers to papers that never appeared for a proof that there exists a $\mathbb{Z}/4$ quotient. For this result (which can be obtained in many ways), I recommend reading the beautiful (but seemingly little-known) paper

Johnson, Dennis and Millson, John J., Modular Lagrangians and the theta multiplier. Invent. Math. 100 (1990), no. 1, 143–165.

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Dr. Putman, you are using Proposition 14 (p. 42) in Dieudonne's book. But couldn't we use instead Lemma 1.1 (p.151) in Johnson & Millson's paper you mention? – Victor Aug 29 '13 at 19:04
@Victor : Yes, but the proof of that proposition uses prop 14 of Dieudonne's book. – Andy Putman Aug 29 '13 at 23:09
Thank you Dr. Putman. One more small question. Where I can find proofs that $\text{Mod}_{g}[2]$ and $\text{Mod}_{g}(q)$ are finite-index subgroups of the mapping class group? – Victor Aug 30 '13 at 2:00
@Victor : It should be obvious to you from the definitions; if not, think about it some more. – Andy Putman Aug 30 '13 at 5:29

In the following paper, explicit finite generating set of the spin mapping class group for the spin structure with Arf invariant 0 was obtained

On diffeomorphisms over surfaces trivially embedded in the 4–sphere, Algebraic & Geometric Topology 2 (2002) 791–824

for the spin structure with Arf invariant 1:

Surfaces in the complex projective plane and their mapping class groups, Algebraic & Geometric Topology 5 (2005) 577–613

I am very happy if the above information is still helpful for you.

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