On Ribbon $R^4$'s

Abstract

We consider ribbon $R^4$'s, that is, smooth open 4-manifolds, homeomorphic to $R^4$ and associated to $h$-cobordisms between closed 4-manifolds. We show that any generalized ribbon $R^4$ associated to a sequence of $h$-cobordisms between non-diffeomorphic 4-manifolds is exotic. Notion of a positive ribbon $R^4$ is defined and we show that a ribbon $R^4$ is positive if and only if it is associated to a sequence of stably non-product h-cobordisms. In particular we show that any positive ribbon $R^4$ is associate to a subsequence of the sequence of non-product h-cobordisms from [BG].