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Register a new userThe Torelli group and congruence subgroups of the mapping class group
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Andrew Putman
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These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class group, following my papers The second rational homology group of the moduli space of curves with level structures and The Picard group of the moduli space of curves with level structures.
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We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,mathbb{Z})$-module.
A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. In the published version of Mapping class group and a global Torelli theorem for hyperkahler manifolds I made an error based on a wrong quotation of Dennis Sullivans famous paper Infinitesimal computations in topology. I claimed that the natural homomorphism from the mapping class group to the group of automorphims of cohomology of a simply connected Kahler manifold has finite kernel. In a recent preprint arXiv:1907.05693, Matthias Kreck and Yang Su produced counterexamples to this statement. Here I correct this error and other related errors, observing that the results of Mapping class group and a global Torelli theorem remain true after an appropriate change of terminology.
For some $g geq 3$, let $Gamma$ be a finite index subgroup of the mapping class group of a genus $g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $Gamma$ should be finite. In this note, we prove two theorems supporting this conjecture. For the first, let $T_x$ denote the Dehn twist about a simple closed curve $x$. For some $n geq 1$, we have $T_x^n in Gamma$. We prove that $T_x^n$ is torsion in the abelianization of $Gamma$. Our second result shows that the abelianization of $Gamma$ is finite if $Gamma$ contains a large chunk (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves. This generalizes work of Hain and Boggi.
We prove that the cohomological dimension of the Torelli group for a closed connected orientable surface of genus g at least 2 is equal to 3g-5. This answers a question of Mess, who proved the lower bound and settled the case of g=2. We also find the cohomological dimension of the Johnson kernel (the subgroup of the Torelli group generated by Dehn twists about separating curves) to be 2g-3. For g at least 2, we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli group in genus 2. The main tool is a new contractible complex, called the complex of cycles, on which the Torelli group acts.
We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, which can be viewed as a mapping class group version of a theorem of Ellenberg-Venkatesh-Westerland about braid groups. These results require studying various simplicial complexes formed by subsurfaces of the surface, generalizing work of Hatcher-Vogtmann.
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