Matthew Horak
Matthew Horak

Ph.D. (2003) Cornell University

First Position
Dissertation
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Abstract: Let $\Sigma$ be a punctured orientable surface with fundamental group isomorphic to the free group $\fn$, and let $\Gamma(\Sigma)$ denote the mapping class group of $\Sigma$. By considering all punctured surfaces and all possible identifications of $\pi_1(\Sigma)$ with $\fn$, we construct a covering of CullerVogtmann Outer space by Teichmüller spaces of punctured surfaces. We prove that the nerve of this cover is contractible, so the action of $\out$ on the nerve gives rise to a spectral sequence converging to the homology of $\out$. The $E^1$ page of this spectral sequence is given by the homology of simplex stabilizers. We prove that the stabilizer of a vertex in the nerve is the mapping class group of a surface, and we identify stabilizers of higherdimensional simplices with stabilizers of sets of conjugacy classes in $\fn$.
We then proceed to examine the $E^\infty$ page of the above spectral sequence. By using Harer's homology stability theorems for mapping class groups to analyze the $d^1$ map, we find a bound on the dimension of the subspace of $H_*(\out;\Q)$ generated by the stable rational mapping class homology. Motivated by the question of whether $\Gamma(\Sigma) \into \out$ is nontrivial on homology, we summarize the constructions of known nontrivial stable homology classes for mapping class groups in terms of three different graph complexes. We then give chain maps between these complexes and several chain complexes that compute the homology of $\out$.