Exotic Mazur manifolds and knot trace invariants

Abstract

From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant ν is an invariant of the smooth 4-manifold associated to a knot K ⊂ S 3 by attaching an n-framed 2-handle to B 4 along K . We also show (modulo forthcoming work of Ozsváth and Szabó) that the concordance invariants τ and are not invariants of such 4-manifolds. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct S 1 × S 2 surgeries, resolving a question from Problem 1.16 in Kirby's list [28].

From zero surgeries to candidates for exotic definite four-manifolds

“…Hence, in this paper we are particularly interested in knots with homeomorphic zero-surgeries which do not have diffeomorphic traces. In full generality, it is a subtle problem to demonstrate that a pair of knot traces with homomorphic boundaries are not diffeomorphic, see [56,27]. In fact, even the easier problem of determining whether given zero surgery homeomorphism can be extended to a trace diffeomorphism is open in general.…”

“…In [56] Yasui gives a construction of knots with homeomorphic 0-surgeries which he uses to disprove the Akbulut-Kirby conjecture. While we enthusiastically recommend his paper, in fact we will model our diagrams off of the (reproduced) proof of his construction given in Section 2.1 of [27].…”

Multisections of 4-manifolds

“…On the other hand, these two 4-sections of S 4 cannot be diffeomorphic, since this would restrict to a diffeomorphism between Z and Z . Applying this construction to the pairs of exotic Mazur manifolds from [HMP19] completes the proof.…”

“…This question is only interesting in the case that the cross-section manifolds are identical; otherwise the multisections could not possibly be diffeomorphic. Our main tool will be the (infinitely many) exotic Mazur manifolds constructed in [HMP19], i.e., pairs of compact contractible 4-manifolds built with a single 1-and 2-handle, that are homeomorphic but not diffeomorphic. Proposition 6.4.…”

Getting a handle on the Conway knot

“…The goal is now to find a way to show that K , the knot that shares a trace with the Conway knot, is not slice. It turns out that some slice obstructions, such as the invariant ν coming from knot Floer homology, are actually trace invariants: if two knots K 1 and K 2 have the same trace, then ν(K 1 ) = ν(K 2 ) [HMP19].…”