Towards an Instanton Floer Homology for Tangles

In this thesis we investigate the problem of defining an extension of sutured instanton Floer homology to give an instanton invariant for a \emph{tangle}. We do this in three separate steps. First, we investigate the representation variety of singular flat connections on a punctured Riemann surface $\Sigma$. Suppose $\Sigma$ has genus $g$ and that there are $n$ punctures. We give formulae for the Betti numbers of the space $\mathcal{R}_{g,n}$ of flat $\SU(2)$-connections on $\Sigma$ with trace 0 holonomy around the punctures. By using a natural extension of the Atiyah-Bott generators for the cohomology ring $H *(\mathcal{R}_{g,n})$, we are able to write down a presentation for this ring in the case $g=0$ of a punctured sphere. This is accomplished by studying the intersections of Poincar\`e dual submanifolds for the new generators and reducing the calculation to a linear algebra problem involving the symplectic volumes of the representation variety.