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The Quarterly Journal of Mathematics Advance Access originally published online on June 30, 2005
The Quarterly Journal of Mathematics 2005 56(3):321-336; doi:10.1093/qmath/hah040
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© The Author 2005. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oupjournals.org

Cut vertices in commutative graphs

James Conant1 *, Ferenc Gerlits2 § and Karen Vogtmann3 ¶

1 1Department of Mathematics, University of Tennessee at Knoxville, Knoxville, TN 37996-1300, USA, 2 2Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13-15, H-1053, Budapest, Hungary, 3 3Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA

The homology of Kontsevich's commutative graph complex parametrizes finite type invariants of odd-dimensional manifolds. This graph homology is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of contact between geometric group theory and quantum topology. In this paper we give two different proofs (one algebraic, one geometric) that the commutative graph complex is quasi-isomorphic to the quotient complex obtained by modding out by graphs with cut vertices. This quotient complex has the advantage of being smaller and hence more practical for computations. In addition, it supports a Lie bialgebra structure coming from a bracket and cobracket we defined in a previous paper. As an application, we compute the rational homology groups of the commutative graph complex up to rank 7.

Received 2 September 2004.


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