Georgia Institute of Technology College of Sciences

In a recent note Francesco Lin showed that if a rational homology sphere Y admits a taut foliation then the Heegaard Floer module HF^-(Y) contains a copy of F[U]/U as a summand. This implies that either the L-space conjecture is false or that Heegaard Floer homology satisfies a geography restriction. In a recent paper in collaboration with Fraser Binns we verified that Lin's geography restriction holds for a wide class of rational homology spheres. I shall discuss our argument, and advance the hypothesis that the Heegaard Floer module HF^-(Y) may satisfy a stronger geography restriction than the one suggested by Lin’s theorem.