Cohomological structure of the mapping class group and beyond

By Shigeyuki Morita. 2005

50 citations recorded by the source corpus.

In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces. Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/39ca9a838a71cbe19b1f78c7e4ce95d6289b3050

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In Geometry and Topology

1. Philippe Elbaz-Vincent
2. Christophe Soulé
3. Herbert Gangl
4. M. Kerz
5. Claire Voisin
6. Don Zagier

In Mathematical Physics

1. Philippe Elbaz-Vincent
2. Christophe Soulé
3. Herbert Gangl
4. M. Kerz

Citations in the corpus

• Perfect forms, K-theory and the cohomology of modular groups (2013, background): “Morita asks in [21] whether the class of infinite order in H(5)(GL5(Z), Z) survives in the cohomology of the group of outer automorphisms of the free group of rank six.”