On the Framed Singular Instanton Floer Homology From Higher Rank Bundles

Abstract

In this thesis we study the framed singular instanton Floer homology defined by by Kronheimer and Mrowka in \cite{KM3}. Given a 3-manifold $Y$ with a link $K$ and $\delta \in H^2(Y,\mathbb{Z})$ satisfying a non-integral condition, they define the singular instanton Floer homology group $I^N(Y,K,\delta)$ by counting singular flat $PSU(N)$-connections with fixed holonomy around $K$. Take a point $x\in Y\backslash K$, classical point class operators $\mu_i (x)$ of degree $2i$ on $I^N(Y,K,\delta)$ can be defined as in the original Floer theory defined by smooth connections. In the singular instanton Floer homology group $I_\ast^N(Y,K,\delta)$, there is a special degree 2 operator $\mu (\sigma)$ for $\sigma \in K$. We study this new operator and obtain a universal relation between this operator and the point class operators $\mu_i (x)$. After restricted to the reduced framed Floer homology $F\bar{I}_\ast^N(Y,K)$, these point classes operators $\mu_i (x)$ become constant numbers related to the $PSU(N)$-Donaldson invariants of four-torus $T^4$. Then the universal relation becomes a characteristic polynomial for the operator $\mu(\sigma)$ so that we can understand the eigenvalues of $\mu(\sigma)$ and decompose the Floer homology as eigenspaces.