Exotic Mazur manifolds and knot trace invariants. (English) Zbl 1482.57019
A Mazur manifold is a compact, contractible \(4\)-manifold with a handle decomposition consisting of a single \(0\)-handle, a single \(1\)-handle, and a single \(2\)-handle. The paper under review constructs the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. This is challenging since most smooth \(4\)-manifold invariants are trivial on manifolds with trivial second homology. The key insight of the paper is to associate knot traces to Mazur manifolds, whereby knot invariants and \(3\)-manifold invariants can be applied to the problem. As a corollary to the existence of exotic Mazur manifolds, integer homology \(3\)-spheres admitting two distinct surgeries to \(S^1 \times S^2\) are produced, answering Problem 1.16 in Kirby’s list.
Reviewer: Arunima Ray (Waltham)
MSC:
| 57K40 | General topology of 4-manifolds |
| 57K10 | Knot theory |
| 57R55 | Differentiable structures in differential topology |
| 57K18 | Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) |
Keywords:
exotic 4-manifolds; Mazur manifolds; knot traces; concordance invariants; knot Floer homologyReferences:
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