Exotic S^2 x S^2 candidates that are just fake

Abstract

We show that two infinite families of homology $S^2\times S^2$'s presented in \cite{FPS, AP}, originally proposed as exotic candidates, are aspherical. In particular, none of these manifolds is an exotic $S^2\times S^2$. We prove this by reconstructing them from aspherical building blocks, following Edmonds. The same arguments apply to and rule out further candidates for simply connected exotica. We also give counterexamples to the conjecture that Luttinger surgery preserves symplectic asphericity.

arXiv comment15 pages. v2 addresses questions received about symplectic asphericity under Luttinger surgery (Theorem 2) and the existence of metrics of strictly negative sectional curvature on these examples (Addendum 13)