arXiv subjects
Publications and source records attributed to Jørgen Ellegaard Andersen.
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.
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 $Σ$.
We show the existence of a universal Vassiliev invariant for links in closed surface cylinders by explicit construction using configuration space integrals.
In this paper we define an algebra structure on the vector space $L(Σ)$ generated by links in the manifold $Σ\times [0,1]$ where $Σ$ is an oriented surface. This algebra has a filtration and the associated graded algebra $L_{Gr}(Σ)$ is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra of chord diagrams $ch(Σ)$ on $Σ$ to $L_{Gr}(Σ)$. We show that multiplication in $L(Σ)$ provides a geometric way to define a deformation quantization of the algebra of chord diagrams, provided there is a universal Vassiliev invariant for links in $Σ\times [0,1]$. The quantization descends to a quantization of the moduli space of flat connections on $Σ$ and it is universal with respect to group homomorphisms. If $Σ$ is compact with free fundamental group we construct a universal Vassiliev invariant.