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Title:
Fixed-point Floer homology in spaces of stable pairs over a Riemann surface
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Abstract:
I will report on joint work with graduate student Andrew Lee, motivated by a proposal that there should exist a “U(2)-Heegaard Floer theory”: an analogue of the Heegaard Floer theory for 3-manifolds, based on Lagrangian Floer theory not in the g-fold symmetric product of the Heegaard surface Σ, but in the moduli space M of rank 2 stable pairs over Σ (as studied by Bradlow and by Thaddeus, among others) of degree 2g+2 and small stability parameter. This is a Fano manifold of dimension 3g; when g=1, it is the blow-up of projective 3-space along a copy of Σ. I expect such a theory will have an Atiyah-Floer type relationship with a gauge-theoretic Floer theory for the 3-manifold, based on a version of the U(2) Seiberg-Witten equations, but so far neither of these theories have been constructed. As an exploratory exercise, we develop a Floer theory for fibered 3-manifolds based on fixed-point Floer homology in M. We compute the resulting groups in genus 1; our argument requires a continuity principle for Floer homology, and a precise knowledge of quantum cohomology for M.
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