The Torelli group and congruence subgroups of the mapping class group

Let Σg,n be a compact oriented genus g surface with n boundary components. The mapping class group of Σg,n, denoted Modg,n, is the group of orientation-preserving diffeomorphisms of Σg,n that restrict to the identity on ∂Σg,n, modulo isotopies that fix ∂Σg,n. The group Modg,n plays a fundamental role in many areas of mathematics, ranging from low-dimensional topology to algebraic geometry. At least for large g, the cohomology of Modg,n is well-understood due to the resolution of the Mumford conjecture by Madsen and Weiss [26]. However, the cohomology of finite-index subgroups of Modg,n remains a mystery. In these notes, we will focus on one low-degree calculation. Consider n ∈ {0, 1}. For an integer p, the level p congruence subgroup of Modg,n, denoted Modg,n(p), is the subgroup of Modg,n consisting of mapping classes that act trivially on H1(Σg,n;Z/p). Another description of Modg,n(p) is as follows. The action of Modg,n on H1(Σg,n;Z) preserves the algebraic intersection pairing. Since n ≤ 1, this is a nondegenerate alternating form, so we obtain a representation Modg,n → Sp2g(Z). Classically this representation was known to be surjective (see §1). Let Sp2g(Z, p) be the subgroup of Sp2g(Z) consisting of matrices which equal the identity modulo p. Then Modg,n(p) is the pullback of Sp2g(Z, p) to Modg,n. These notes will discuss the calculation of H(Modg,n(p);Z). One motivation for this is the study of line bundles on the finite cover of the moduli space of curves associated to Modg,n(p), which is known as the moduli space of curves with level p structures. The first Chern class of such a line bundle lies in H(Modg,n(p);Z), and the determination of H(Modg,n(p);Z) is the heart of the paper [35], which gives a ∗Supported in part by NSF grant DMS-1005318