Towards an Instanton Floer Homology for Tangles

Abstract

In this thesis we investigate the problem of defining an extension of sutured instanton Floer homology to give an instanton invariant for a 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 Poincaré dual submanifolds for the new generators and reducing the calculation to a linear algebra problem involving the symplectic volumes of the representation variety. We then study the related problem of computing the instanton Floer homology for a product link in a product 3-manifold <p>\((Y_g, K_n) := (S^1 \times \Sigma, S^1 \times \{n pts\})\).<\p> It is easy to see that the Floer homology of this pair, as a vector space, is essentially the same as the cohomology of \(\mathcal{R}_{g,n}\), and so we set ourselves to determining a presentation for the natural algebra structure on it in the case \(g = 0\). By leveraging a stable parabolic bundles calculation for \(n = 3\) and an easier version of this Floer homology, \(I _*(Y_0, K_n, u)\), we are able to write down a complete presentation for the Floer homology \(I _*(Y_0, K_n)\) as a ring. We recapitulate somewhat the techniques in \([\boldsymbol{27}]\) in order to do this. Crucially, we deduce that the eigenspace for the top eigenvalue for a natural operator \(\mu^{ orb} (\Sigma)\) on \(I_* (Y_0, K_n)\) is 1-dimensional.Finally, we leverage this 1-dimensional eigenspace to define an instanton tangle invariant THI and several variants by mimicking the de nition of sutured Floer homology SHI in \([\boldsymbol{22}]\). We then prove this invariant enjoys nice properties with respect to concatenation, and prove a nontriviality result which shows that it detects the product tangle in certain cases.