The Johnson homomorphism and its generalizations (Richard Hain, Duke University)

05.04.2019 10:30 – 12:30

The Torelli group of a compact orientable surface is the subgroup of its mapping class group that consist of those isotopy classes of diffeomorphisms that act trivially on the homology of the associated closed surface. Torelli groups are important in topology (the study of homology 3-spheres) and algebraic geometry (their connections to the Ceresa cycle and tautological rings of moduli spaces of curves).
The Johnson homomorphism and its generalizations are tools for understanding Torelli groups. In this talk I will survey the work of many mathematicians (Morita, Enomoto, Sato, ...) on generalizations of Johnson's original homomorphism, including recent work of Kawazumi and Kuno on the completed Goldman--Turaev Lie bialgebra and its relation to computing the images of generalized Johnson homomorphisms.
Hodge theory (equivalently, Galois actions) provides additional structure that can be used to understand the Johnson homomorphism and its image. This leads to the definition of "arithmetic Johnson homomorphisms". I will explain how Oda's conjecture (proved by Takao), the work of Francis Brown, and the Turaev cobracket combine to constrain the image of the arithmetic Johnson homomorphism.

Lieu

Bâtiment: Villa Battelle

Séminaire "Groupes de Lie et espaces des modules"

entrée libre