Computations in formal symplectic geometry and characteristic classes of moduli spaces

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.

33 pages, final version, to appear in Quantum Topology

References in corpus (7)

• The Gelfand-Kalinin-Fuks class and characteristic classes of transversely symplectic foliations
• A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)
• Lie algebras of symplectic derivations and cycles on the moduli spaces
• Structure of symplectic invariant Lie subalgebras of symplectic derivation Lie algebras
• Canonical metric on the space of symplectic invariant tensors and its applications
• On the S_n-equivariant Euler characteristic of M_{2,n}
• Symmetry of symplectic derivation Lie algebras of free Lie algebras

Cited by in corpus (7)

• Tropical curves, graph complexes, and top weight cohomology of M_g
• Assembling homology classes in automorphism groups of free groups
• A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)
• Tropical curves and covers and their moduli spaces
• Structure of symplectic invariant Lie subalgebras of symplectic derivation Lie algebras
• Canonical metric on the space of symplectic invariant tensors and its applications
• Secondary characteristic classes for subgroups of automorphism groups of free groups