122 research outputs found
Lie algebras of symplectic derivations and cycles on the moduli spaces
We consider the Lie algebra consisting of all derivations on the free
associative algebra, generated by the first homology group of a closed oriented
surface, which kill the symplectic class. We find the first non-trivial
abelianization of this Lie algebra and discuss its relation to unstable
cohomology classes of the moduli space of curves via a theorem of Kontsevich.Comment: This is the version published by Geometry & Topology Monographs on 25
February 200
Computations in formal symplectic geometry and characteristic classes of moduli spaces
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.Comment: 33 pages, final version, to appear in Quantum Topolog
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