The defining properties of the Kontsevich unoriented graph complex

Abstract

Consider the real vector space of formal sums of non-empty, finite unoriented graphs without multiple edges and loops. Let the vertices of graphs be unlabelled but let every graph $\gamma$ be endowed with an ordered set of edges $\mathsf{E}(\gamma)$. Denote by Gra the vector space of formal sums of graphs modulo the relation $(\gamma_1,\mathsf{E}(\gamma_1))-\text{sign}(\sigma) (\gamma_2,\mathsf{E}(\gamma_2)) = 0$ for topologically equal graphs $\gamma_1$ and $\gamma_2$ whose edge orderings differ by a permutation $\sigma$. The zero class in Gra is represented by sums of graphs that cancel via the above relation. The Lie bracket of graphs with ordered edge sets is defined using the insertion of a graph into vertices of the other one. We give an explicit proof of the theorems which state that the space Gra is a well\/-\/defined differential graded Lie algebra: both the Lie bracket $[{\cdot},{\cdot}]$ and the vertex\/-\/expanding differential ${\mathrm d}=[{\bullet}\!{-}\!{\bullet},{\cdot}]$ respect the calculus modulo zero graphs.