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Graph homology and configuration spaces
by
Vladimir Baranovsky
University of California - Irvine
Coauthors: Radmila Sazdanovic
We will discuss the proof of the conjecture due to M. Khovanov relating the chromatic graph homology over algebra A defined by L. Helme-Guizon and Y. Rong, and the homology of a graph configuration space of a compact oriented manifold M which has cohomology algebra A. The latter construction was introduced by M. Eastwood, S. Huggett.
We show that there is a spectral sequence with the E_1 term given by the graph homology of A, and converging to the homology of the graph configuration space. When the graph is the complete graph on n vertices, the statement is essentially due to Bendersky and Gitler. In the same setting Totaro and others have proved that the spectral sequence degenerates if M is Kahler, or formal.
In their original work, Bendersky and Gitler have also conjectured that the differentials of the spectral sequence are, in some sense, given by the Massey products of M, and we prove this conjecture in the more general graph setting (earlier it was known for a complete graph on n=4 vertices, due to Felix and Thomas).
We will conclude by formulating two open questions related to the spectral sequence.
Date received: October 31, 2011
Copyright © 2011 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbdt-12.