Contact Manifolds with Flexible Fillings

We prove that all flexible Weinstein fillings of a given contact manifold with vanishing first Chern class have isomorphic integral cohomology. As an application, we show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. We also construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. These contact structures are distinguished by positive symplectic homology, which we prove is a contact invariant for flexibly-filled contact structures. The key step is a procedure for increasing the degrees of Reeb chords of loose Legendrians.