Cut vertices in commutative graphs

Details Cut vertices in commutative graphs Cut vertices in commutative graphs

2003-07-28

arxiv.org/abs/math/0307364v3

Q. J. Math. 56 (2005), no. 3, 321--336

Mathematics

Quantum Algebra

Final version. To appear in Oxford Quarterly Journal. Programs and data are available with the version one source of this pape

Scientific paper

The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it 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.

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