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New connections in low-dimensional topology: Relating Heegaard Floer homology and the fundamental group

Final Report Summary - HFFUNDGRP (New connections in low-dimensional topology: Relating Heegaard Floer homology and the fundamental group)

The main project goals were achieved. In particular a complete solution to the L-space conjecture for graph manifolds was achieved , and was published in the paper "Taut foliation on graph manifolds", joint with J. Hanselman, J. Rasmussen and S. Rasmussen, arXiv1508.05911. This established the most general currently known equivalence between left orderable fundamental groups, taut foliations on 3-manifolds and non-L-spaces. Two follow-up papers carried the Foer-theoretic machinery developed in the above paper further. Taken together, this work constitutes a step change in understanding of Heegard Floer homology for manifolds with torus boundary, fulfilling the objectives of the project.