Seifert-fibred homology 3-spheres, V -surfaces and the Floer index

Source abstract

Let Σ be a Seifert-fibred homology 3-sphere. We are interested in the chain complex for the Floer homology of Σ. This is generated by the critical points of the ChernSimons functional acting on the moduli space of irreducible SU (2)-connections modulo gauge-equivalence, i.e. the equivalence classes of flat connections: see [6]. Specifically, we ask the question: given the holonomy ρ of a flat connection C ρ , what is the index of C ρ in the chain complex? By definition this Floer index is given by the spectral flow of a family of twisted signature operators with coefficients in the adjoint bundle (with fibre su(2)) corresponding to any path of connections joining the trivial connection to C ρ . We will calculate this spectral flow by an almost entirely direct method, obtaining a formula in terms of the dimensions of spaces of π 1 (Σ)- automorphic functions. These dimensions will be evaluated to give the numerical result previously obtained using a different method by Fintushel and Stern: see [5].