A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain complex whose underlying vector space is freely generated by isomorphism classes of oriented ribbon graphs and the differential is given contracting edges (which are not loops) in all possible ways. It is well known that the homology of this complex is related to the homology of the moduli space of Riemann surfaces with decorated punctures.

In the 1990's Kontsevich described graph complexes in a purely algebraic manner by relating the homology of an infinite dimensional Lie algebra to the homology of the ribbon graph complex. More precisely, the standard symplectic form on $V=\mathbb{R}^{2n}$ can be extended to a Lie bracket on the vector space of cyclic polynomials functions $\mathfrak{g}_n=\bigoplus_{i=2}^{\infty} (V^*)^{\otimes i}_{\mathbb{Z}/i\mathbb{Z}}$ and we can construct a chain map $\psi_n: C_*(\mathfrak{g}_n) \to \mathcal{G}_*$ where $C_*(\mathfrak{g}_n)$ is the Chevalley Eilenberg complex of $\mathfrak{g}_n$. The result is that if we take the direct limit with respect to $n$, the induced map $\psi_{\infty}$ induces an isomorphism on homology. The proof uses a clever application of a fact from the invariant theory of the symplectic Lie algebra. The theorem generalizes to all cyclic operads.

In what sense is this infinite dimensional Lie algebra better than the ribbon graph complex? Is there anything we can do with it that we can not do with graphs? Perhaps it facilitates calculations, construction of non trivial homology classes, or constructions of 2d TQFT's?


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