The dimensions and Euler characteristics of M. Kontsevich's graph complexes

Abstract

We provide a generating function for the (graded) dimensions of M. Kontsevich's graph complexes of ordinary graphs. This generating function can be used to compute the Euler characteristic in each loop order. Furthermore, we show that graphs with multiple edges can be omitted from these graph complexes.

Integral Euler characteristic of Out F_{11}

“…Finally, we comment about related works on the other two cases of Kontsevich's theorem. As for the commutative case, Willwacher and Živković [17] recently obtained the generating function of the (total) Euler characteristic and computed the explicit values up to weight 60. Our former results in [13] are consistent with theirs.…”