From zero surgeries to candidates for exotic definite four-manifolds

Abstract

One strategy for distinguishing smooth structures on closed $4$-manifolds is to produce a knot $K$ in $S^3$ that is slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic smooth filling $W^{\prime }$. In this paper we explore how $0$-surgery homeomorphisms can be used to potentially construct exotic pairs of this form. In order to systematically generate a plethora of candidates for exotic pairs, we give a fully general construction of pairs of knots with the same zero surgeries. By computer experimentation, we find $5$ topologically slice knots such that, if any of them were slice, we would obtain an exotic four-sphere. We also investigate the possibility of constructing exotic smooth structures on ${\#}^n\mathbb{C}{P}^2$ in a similar fashion.

arXiv commentfinal version, to appear in J. London Math. Soc.; supporting files found at http://web.stanford.edu/~cm5/RBG.html

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