Cappell–Shaneson's $4$–dimensional $s$–cobordism

Abstract

In 1987 S Cappell and J Shaneson constructed an $s$–cobordism $H$ from the quaternionic 3–manifold $Q$ to itself, and asked whether $H$ or any of its covers are trivial product cobordism? In this paper we study $H$, and in particular show that its 8–fold cover is the product cobordism from $S^3$ to itself. We reduce the triviality of $H$ to a question about the 3–twist spun trefoil knot in $S^4$, and also relate this to a question about a Fintushel–Stern knot surgery.