Distortion of Mappings and $L_{q,p}$-Cohomology

The first cohomology of the mapping class group with coefficients in algebraic functions on the SL(2, C) moduli space

Consider a compact surface of genus at least two. We prove that the first cohomology group of the mapping class group with coefficients in the space of algebraic functions on the SL(2, C) moduli space vanishes.

Degree one cohomology with twisted coefficients of the mapping class group

Let $Γ$ be the mapping class group of an oriented surface $Σ$ of genus g with r boundary components. We prove that the first cohomology group $H^1(Γ, O(M_{SL(2, C)})^*)$ is non-trivial, where the coefficient module is the dual of the space of algebraic functions on the $SL(2, C)$ moduli space over $Σ$.

Cohomology of mapping class groups and the abelian moduli space

We consider a surface $Σ$ of genus $g \geq 3$, either closed or with exactly one puncture. The mapping class group $Γ$ of $Σ$ acts symplectically on the abelian moduli space $M = \Hom(π_1(Σ), U(1)) = \Hom(H_1(Σ),U(1))$, and hence both $L^2(M)$ and $C^\infty(M)$ are modules over $Γ$. In this paper, we prove that both the cohomology groups $H^1(Γ, L^2(M))$ and $H^1(Γ, C^\infty(M))$ vanish.

continious cohomology of groups of volume-preserving and symplectic diffepmorphisms, measurable transfer and higher asymtotic cycles

I construct the real counterparts (which I call Borel-Bott classes) of the R/Z classes constructed in "Characteristic classes in symplectic topology", to appear, in the cohomology of volume-preserving and symplectomorhisms of a compact (symplectic) manifold.I show that, for the symplectic action of the mapping class group in the moduli space of stable vector bundles over a Riemann surface, the restriction of the first constructed class from the symplectomorphism group gives a generator for the second (bounded) cohomology of the mapping class group.

度量扭曲、外微分与高斯引理

Metrical Distortion, Exterior Differential and Gauss's Lemma

本文修正了高斯引理，提出度量扭曲的概念，并通过协变梯度运输定义外微分，结合测地径向体积与长度保持，示例说明2球面的外几何特性。

Equivariant Poisson Cohomology and a Spectral Sequence Associated with a Moment Map

We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the $E_2$-term equal to the tensor product of the cohomology of the Lie algebra and the equivariant Poisson cohomology of the manifold. The latter is defined as the equivariant cohomology of the multi-vector fields made into a G-differential complex by means of the momentum mapping.…