Stable cohomology of graph complexes

Abstract

We study three graph complexes related to the higher genus Grothendieck–Teichmüller Lie algebra and diffeomorphism groups of manifolds. We show how the cohomology of these graph complexes is related, and we compute the cohomology as the genus g tends to $$\infty $$ ∞ . As a byproduct, we find that the Malcev completion of the genus g mapping class group relative to the symplectic group is Koszul in the stable limit, partially answering a question of Hain.

On the Torelli Lie algebra

“…We also prove that Gr • LCS 𝔱 𝑔 and Gr • LCS 𝔱 1 𝑔 are Koszul in the same range, as well as the relative unipotent completions Gr • LCS 𝔲 𝑔 , Gr • LCS 𝔲 1 𝑔 and Gr • LCS 𝔲 𝑔,1 (see Section 3.1.1 for detailed definitions). This result for Gr • LCS 𝔲 𝑔 has simultaneously been obtained by Felder-Naef-Willwacher [FNW23]; we discuss the relation between the arguments in Remark 7.4.…”

“…Up to a change of degrees the chain complex may be identified with the sum of the graph complexes studied by Chan–Galatius–Payne [CGP21, CGP22]. When , this is predual to Kontsevich’s graph complex as studied by Willwacher [Wil15], and for general S it is predual to the ‘hairy graph complex’ of Andersson–Willwacher–Zivkovic [AWZ20].…”

“…When , this is predual to Kontsevich’s graph complex as studied by Willwacher [Wil15], and for general S it is predual to the ‘hairy graph complex’ of Andersson–Willwacher–Zivkovic [AWZ20]. Similarly, the chain complex may be identified with the sum of the reduced chains on the moduli spaces of tropical curves of genus g with marked points from [CGP22], or the predual of from [AWZ20].…”

“…In the degree convention of [CGP22], a connected (weighted) graph is given degree , where is the sum of the first Betti number of and the weights at all its vertices. Our internal degree can be compared to via and our total degree is , so …”

“…Willwacher has pointed out the following line of reasoning to us, which is essentially the argument of [FNW23]: Combining the above with the discussion of Chan–Galatius–Payne’s complexes in the last section, it follows that Koszulness of is equivalent to for , and Koszulness of is equivalent to for , but both follow from [CGP22, Theorem 1.6].…”