Traces of links and simply connected 4-manifolds

Abstract

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