Computations in formal symplectic geometry and characteristic classes of moduli spaces

TL;DR

This paper performs explicit computations in formal symplectic geometry to determine Euler characteristics of various algebraic structures, revealing new cohomology classes and proposing conjectures on graph homology and moduli space characteristics.

Contribution

It provides explicit Euler characteristic calculations for commutative, Lie, and associative cases, and explores their implications for automorphism groups and moduli spaces.

Findings

Euler characteristics of Out F_n for n ≤ 10 are computed.

Existence of rational cohomology classes of odd degrees is established.

New non-trivalent graph homology classes of odd degrees are identified.

Abstract

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups Out F_n of free groups for all n <= 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new non-trivalent graph homology classes of odd degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves.