On homology of Torelli groups

Alexander Gaifullin, Steklov Mathematical Institute of Russian Academy of Sciences & Skolkovo Institute of Science and Technology

Zoom link: https://princeton.zoom.us/j/96282936122

Passcode: 998749

Theory of mapping class groups of oriented surfaces is closely related to geometry and topology of moduli spaces, topology of three-dimensional manifolds, and automorphisms of free groups. One of the most important subgroups of the genus $g$ mapping class group is the Torelli group $I_g$ consisting of all mapping classes that act trivially on homology of the surface. Interest in its study is caused, among other things, by two facts. Firstly, $I_g$ is the fundamental group of the Torelli space, which can be regarded as the moduli space of genus $g$ smooth complex curves with a fixed symplectic basis of the first homology group. Secondly, elements of the Torelli groups provide Heegaard splittings of three-dimensional homology spheres. These facts yield in a fruitful interplay of methods of different nature in the study of Torelli groups.

Though $I_g$ has finite cohomological dimension $3g-5$ (Bestvina-Bux-Margalit, 2007), it is known that the Eilenberg-MacLane space $K(I_g,1)$ is not homotopy equivalent to a finite CW-complex, provided that $g>1$. One of interesting open questions concerning Torelli groups is the question of starting from which $k$ the topology of $K(I_g,1)$ becomes infinite. More precisely, for which largest $k$ the space $K(I_g,1)$ is homotopy equivalent to a CW-complex with finite $(k-1)$-skeleton. This question is closely related to the question of finding the smallest $k$ for which the group $H_k(I_g)$ is not finitely generated. In 1980 Johnson proved that, for $g\ge 3$, the Torelli group $I_g$ is finitely generated and hence $k>2$ for both questions.

In the talk I will focus on my recent result claiming that the group $H_k(I_g)$ is not finitely generated for each $g$ satisfying $2g-3\le k < 3g-5$. (For $k=3g-5$, this was previously known by a result of Bestvina, Bux, and Margalit). Also I will present a partial result providing evidence for the conjecture that $H_2(I_3)$ is not finitely generated and hence $I_3$ is not finitely presented. The main tool is the study of differentials of the spectral sequence associated with the action of $I_g$ on the complex of cycles constructed by Bestvina, Bux, and Margalit.