On Ribbon $R^4$'s

TL;DR

This paper investigates the properties of ribbon R^4's, showing that generalized ribbon R^4's linked to non-diffeomorphic 4-manifolds are exotic, and characterizes positive ribbon R^4's via stably non-product h-cobordisms.

Contribution

It introduces the notion of positive ribbon R^4 and characterizes it in terms of sequences of stably non-product h-cobordisms, linking to previous work.

Findings

Generalized ribbon R^4's associated with non-diffeomorphic manifolds are exotic.

Positive ribbon R^4's correspond to sequences of stably non-product h-cobordisms.

Any positive ribbon R^4 is related to a subsequence of known non-product h-cobordisms.

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].