Heegaard Floer homology and the fundamental group

By the geometrization theorem, the fundamental group determines an irreducible three-manifold except lens spaces. It follows that the Heegaard Floer homology of a three-manifold (or a null homologous knot) is determined by its fundamental group. A direct relationship between the fundamental group and Heegaard Floer homology is expected. We show that the hat version knot Floer homology of a (1,1) knot is determined by certain presentations of its fundamental group. This is joint work with Matthew Hedden and Xiliu Yang.

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