Around the L-space conjecture
The L-space conjecture asserts that for (closed, orientable, prime) three-manifolds, left-orderability of the fundamental group, the existence of a coorientable taut foliation, and admitting non-trivial Heegaard Floer homology are equivalent conditions. In particular, this suggests two topological characterizations of those manifolds with simplest-possible Heegaard Floer homology — these are called L-spaces. More interesting, this conjecture ties together structures arising in low-dimensional topology in a way that is somewhat surprising. This talk will introduce the different moving parts in the conjecture and survey some of what is known, with a particular focus on recent developments in bordered Heegaard Floer theory leading to, among other things, establishing the L-space conjecture for all graph manifolds.