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

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.