Traces of links and simply connected 4‐manifolds

Abstract

We study the set S ̂ M $\widehat{\mathcal {S}}_M$ of framed smoothly slice links which lie on the boundary of the complement of a 1-handlebody in a closed, simply connected, smooth 4-manifold M $M$ . We show that S ̂ M $\widehat{\mathcal {S}}_M$ is well defined and describe how it relates to exotic phenomena in dimension four. In particular, in the case when X $X$ is a smooth 4-manifold-with-boundary, with a handle decompositions with no 1-handles and homeomorphic to but not smoothly embeddable in D 4 $D^4$ , our results tell us that X $X$ is exotic if and only if there is a link L ↪ S 3 $L\hookrightarrow S^3$ which is smoothly slice in X $X$ , but not in D 4 $D^4$ . Furthermore, we extend the notion of high genus 2-handles attachment, introduced by Hayden and Piccirillo, to prove that exotic 4-disks that are smoothly embeddable in D 4 $D^4$ , and therefore possible counterexamples to the smooth 4-dimensional Schönflies conjecture, cannot be distinguished from D 4 $D^4$ only by comparing the slice genus functions of links.