The Torelli group and congruence subgroups of the mapping class group (Q2848315)

Let \(S_{g,n}\) be a compact oriented surface of genus \(g\) with \(n\) boundary components. Denote by \(\text{Mod}_{g,n}\) the mapping class group of \(S_{g,n}\), i.e., the group of orientation-preserving diffeomorphisms of \(S_{g,n}\) that restrict to the identity on the boundary \(\partial S_{g,n}\), modulo isotopies that fix \(\partial S_{g,n}\). The present set of lecture notes is devoted to the study of certain finite-index subgroups of \(\text{Mod}_{g,n}\) and their (co-)homological properties. Actually, the author considers integers \(n\in\{0,1\}\) and \(p\geq 0\) together with the corresponding level \(p\) congruence subgroup of \(\text{Mod}_{g,n}\), denoted by \(\text{Mod}_{g,n}(p)\) where the latter is defined as the subgroup of \(M_{g,n}\) consisting of mapping classes that act trivially on the homology group \(H_1(S_{g,n};\mathbb{Z}/p)\).NEWLINENEWLINE In this context, the main goal of these notes is to discuss the calculation of \(H^2(\text{Mod}_{g,n}(p); \mathbb{Z})\) which is motivated by the study of line bundles on the finite cover of the moduli space \(M_{g,n}\) associated to the subgroup \(M_{g,n}(p)\), that is, on the moduli space of \((g,n)\)-curves with level \(p\) structures. While this study was untertaken in the author's paper [Duke Math. J. 161, No. 4, 623--674 (2012; Zbl 1241.30015)], the present discussion uses this computation for further interesting questions related to the mapping class group. More precisely, the author derives an exact sequence of the form NEWLINE\[NEWLINE0\to \text{Ext}(H_1(\text{Mod}_{g,n}(p); \mathbb{Z}),\mathbb{Z})\to H^2(\text{Mod}_{g,n}(p); \mathbb{Z})\to\Hom(H_2(\text{Mod}_{g,n}(p); \mathbb{Z},\mathbb{Z})\to 0NEWLINE\]NEWLINE and computes the kernel (in Lecture 3) and the cokernel (in Lecture 4) of this sequence explicitely.NEWLINENEWLINE Lecture 1 gives a brief introduction to the Torelli subgroup \(I_{g,n}\) of \(\text{Mod}_{g,n}\) for \(n\in\{0,1\}\), thereby introducing the reader to the related fundamental works of \textit{J. S. Birman}, \textit{D. Johnson} and the author [Geom. Topol. 11, 829--865 (2007; Zbl 1157.57010)] on this subject. Lecture 2 explains the so-called Johnson homomorphism \(I_{g,1}\to H_1(S_{g,0}; \mathbb{Z})\) via two different approaches. These ingredients are then used to calculate the homology groups \(H_1(\text{Mod}_{g,1}(p);\mathbb{Z})\) and \(H_2(\text{Mod}_{g, 0}(p);\mathbb{Q})\) in the remaining two lectures, respectively. Further crucial tools for these calculations are the abelianization of the symplectic group \(\mathrm{Sp}(\mathbb{Z},p)\) and some homological stability methods. As for the second rational homology group of the level \(p\) congruence subgroup \(\text{Mod}_{g,0}(p)\), which is computed in the concluding Lecture 4, the main reference is the author's recent paper [Adv. Math. 229, No. 2, 1205--1234 (2012; Zbl 1250.14019)]. In the course of these very enlightening lectures, the reader is invited to work a large number of related exercises in order to complement the material in a self-reliant, creative way.NEWLINENEWLINEFor the entire collection see [Zbl 1272.30002].