On 3-manifolds that are boundaries of exotic 4-manifolds

TL;DR

This paper establishes criteria for 3-manifolds to bound exotic 4-manifolds with infinitely many smooth structures, especially for contact manifolds with certain invariants, revealing rich boundary-smooth structure relationships.

Contribution

It provides new criteria linking 3-manifold properties to the existence of exotic smooth structures on bounding 4-manifolds, including for contact manifolds with specific invariants.

Findings

Certain 3-manifolds bound infinitely many smooth structures on 4-manifolds.

Weakly fillable contact 3-manifolds admit exotic caps with symplectic structures.

Contact 3-manifolds with non-vanishing Heegaard Floer invariants bound exotic smooth 4-manifolds.

Abstract

We give several criteria on a closed, oriented 3-manifold that will imply that it is the boundary of a (simply connected) 4-manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3-manifold, or contact 3-manifolds with non-vanishing Heegaard Floer invariant, is the boundary of a simply connected 4-manifolds that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any such contact manifold.

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