XXVI International Fall Workshop on Geometry and Physics

Braga, 4-7 September 2017

Heegaard Floer homology is a 3-manifold invariant that computes the dimensional reduction of Seiberg-Witten gauge theory invariants for 4-manifolds. Especially in the case of 3-manifolds with vanishing first cohomology, however, it is still poorly understood what Heegaard Floer homology tells us about a 3-manifold, in terms of other 3-manifold properties or structures that one might want to study.

Recent attempts to solve this mystery involve conjectures relating the occurrence of certain structures---such as taut foliations on the manifold or a left-multiplication-invariant order on the fundamental group of the manifold---to the nonvanishing of the reduced Heegaard Floer homology. I will discuss recent progress in this endeavor, along with potential avenues for finding a more intrinsic interpretation of reduced Heegaard Floer classes.

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