# Towards an Instanton Floer Homology for Tangles

 Title: Towards an Instanton Floer Homology for Tangles Author: Street, Ethan J. Citation: Street, Ethan J. 2012. Towards an Instanton Floer Homology for Tangles. Doctoral dissertation, Harvard University. Full Text & Related Files: Street_gsas.harvard_0084L_10193.pdf (1.774Mb; PDF) 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

$$(Y_g, K_n) := (S^1 \times \Sigma, S^1 \times \{n pts\})$$.<\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. Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:9396423 Downloads of this work: