Mercredi 10 septembre 2014
IRMA
This special day on « Mapping class groups of surfaces and automorphism groups of free groups » is organized in the framework of a joint seminar CNRS-JSPS. _ There will be four research talks, which will be followed by informal discussions on the subject. _ _ If you are planning to attend this meeting, please send an e-mail to G. Massuyeau. _ _ N.B. The seminars will take place in the Conference Room of the IRMA.
-
Mercredi 10 septembre 2014
-
09:30
Nariya Kawazumi, University of Tokyo
The Turaev cobracket, the Enomoto-Satoh traces and the divergence cocycle in the Kashiwara-Vergne problem
We introduce a regular homotopy version of the Turaev cobracket, and interpret the Enomoto-Satoh obstructions for the surjectivity of the Johnson homomorphisms and the divergence cocycle in the Kashiwara-Vergne problem as some part of the `regular' Turaev cobracket. -
11:00
Christine Vespa, University of Strasbourg
Stable homology of automorphism groups of free groups via functor homology
Recently Galatius proved that stable homology of automorphism groups of free groups with integer coefficients is isomorphic to the stable homology of symmetric groups. For twisted coefficients, Satoh obtained several results for the homological degree 1 and 2, for coefficients given by the abelianization functor, the stable homology is trivial by a work of Hatcher-Wahl and Randal-Williams remarked that similar methods give the triviality of stable homology for any tensor power of abelianization. In a joint work with Aurélien Djament, we prove the following general result: stable homology of automorphism groups of free groups with coefficients given by polynomial covariant functors on free groups (for example, any tensor, symmetric or exterior power of the abelianization) is trivial. This work is based on a general method to compute stable group homology with twisted coefficients using functor homology. In this talk, I will explain this general method and develop the particular case of automorphism groups of free groups. -
14:00
Takuya Sakasai, University of Tokyo
Structure of the symplectic derivation Lie algebra of a free Lie algebra
The (stable) symplectic derivations of a free Lie algebra form a Lie algebra. It plays a fundamental role in the theory of Johnson homomorphisms and also appears as one of the three infinite-dimensional Lie algebras considered by Kontsevich in his theory of formal symplectic geometry. We investigate the structure of this Lie algebra by using the representation theory of the symplectic group. More specifically, we discuss its symplectic invariant part and abelian quotients together with their applications to topology. This is joint work with Shigeyuki Morita and Masaaki Suzuki. -
15:30
Yusuke Kuno, Tsuda College, Tokyo
The Morita-Penner construction and the Earle class
The fatgraph complex associated to an orientable surface is a combinatorial tool to study the Teichmuller space and the mapping class group action on it. In 2008, Morita and Penner gave a 1-cocycle on the fatgraph complex with values in the third exterior power of the first homology of the surface. This provides an explicit cocycle for the extended first Johnson homomorphism. By contracting coefficients, we also get an explicit cocycle for the Earle class. The Earle class is a generator of the first cohomology of the mapping class group with coefficients in the first homology of the surface. In this talk, we give a more simple 1-cocycle on the fatgraph complex providing a cocycle for the Earle class. This is based on a joint work with Robert Penner and Vladimir Turaev.