Ferenc Gerlits
Ferenc Gerlits

Ph.D. (2002) Cornell University

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Abstract: We study the homology of various graph complexes. These are chain complexes where the chain groups are spanned by a finite set of graphs.
Graph complexes were first used to compute the homology of mapping class groups. The group of outer automorphisms of the free group $\Out(F_n)$, and the group of automorphisms of the free group $\Aut(F_n)$ are similar to the mapping class groups in many ways. In particular, their homology can be computed using a similar graph complex.
In Chapter 2, we compute $H_*(\Out(F_n); Q)$ for $n\le 5$ using the cell decomposition of Culler and Vogtmann. Hatcher constructed a graph complex to compute the rational homology of \autfn. Hatcher and Vogtmann simplified the graph complex and computed $H_i(\Aut(F_n); Q)$ for $i \le 6$ (and all n). In Chapter 3, we use new algorithms to compute the homology of their graph complex. Our results confirm their computation, and extend it one step further: we compute $H_7(\Aut(F_5); Q) \isomorphic Q$. This is interesting, because it is the first known case where the homology of $\Out(F_n)$ and $\Aut(F_n)$ are different, thus establishing a lower bound for the stability range of the map $H_*(\Aut(F_n)) \to H_*(\Out(F_n))$, which was shown to be an isomorphism for large n by Hatcher.
In Chapter 4, we consider a family of graph complexes introduced by Kontsevich in the study of "noncommutative symplectic geometry.'' He showed that the homology of the Lie algebra of certain symplectic vector fields on R^{2n}, and of noncommutative analogs of this Lie algebra, can be computed by a fairly simple graph complex. We give a short summary of the proof of this theorem, and compute the homology in the commutative case in low dimensions.
Chapter 5 contains a proof of Kontsevich's formula for the Euler characteristic of his graph complexes. This is an application of the method of Feynman diagrams from quantum physics, combined with the combinatorics of species developed by A. Joyal.