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?**



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


