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Mapping class groups and homeomorphism groups are symmetry groups of surfaces and all manifolds. They are usually very rigid, in the sense that their automorphism groups only consist of conjugation. I...
These notes are based on lectures given at Zhejiang University, July 14–20, 2008. My goal was to present some analytic techniques that can be used to study the action of the mapping class group on the...
Let be a compact orientable surface with genus g andnboundary components ∂1,..., ∂n. Let b = (b1,...,bn) ∈ [−2,2] n. Then the mapping class group Mod() acts on the relative SU(2)-...
When G is a connected compact Lie group, and π is a closed surface group, then Hom(π, G)/G contains an open dense Out(π)-invariant subset which is a smooth symplectic manifold.
Deformation spaces Hom(π, G)/G of representations of the fundamental group π of a surface Σ in a Lie group G admit natural actions of the mapping class group ModΣ, preserving a Poisson structure. Wh...
We derive two types of linearity conditions for mapping class groups of orientable surfaces; one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case...
We show that the action of the mapping class group on bordered Floer homology in the second to extremal spinc-structure is faithful. The paper is self-contained, and in particular does not assume any ...
The Torelli group, I(Sg), is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface.
Let g,1 be a compact oriented surface of genus g with one boundary component,andMg,1 its mapping class group. Morita showed that the image of the k-th Johnson homomorphism M k of Mg,1 is contained ...
Given a Lagrangian sphere in a symplectic 4-manifold (M, ω) with b+ = 1, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension of (M, ω) is −∞, this result t...
We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We...
We compute the stable cohomology of the universal Picard stack Pic g ! M g, and also its Picard group. The degree zero Picard stack Pic0 g has homotopy type the classifying space of Kawazumi’s extende...
Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\geq 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. O...
Given a closed, oriented surface , possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of xed points of a surface symplectomorphism (i.e. area-preserving map) in ...
In this survey paper, we give a complete list of known results on the first and the second homology groups of surface mapping class groups. Some known results on higher (co) homology are also mentione...

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