Is the geography of Heegaard Floer homology restricted or the $L$-space conjecture false?

Antonio Alfieri, Fraser Binns

In a recent note F. Lin showed that if a rational homology sphere $Y$ admits a taut foliation then the Heegaard Floer module $H{F}^{-}(Y)$ contains a copy of $\mathbf{F}[U]/U$ as a sum…