Distinguishing closed 4-manifolds by slicing

TL;DR

This paper presents the first examples of distinguishing closed 4-manifolds using slicing knots, including new manifolds with nonvanishing Seiberg-Witten invariants and a novel construction related to complex projective planes.

Contribution

It provides the first successful examples of differentiating 4-manifolds via slicing knots and introduces new manifolds with specific invariants and topological properties.

Findings

First examples of 4-manifolds distinguished by slicing knots.

Construction of 4-manifolds with nonvanishing Seiberg-Witten invariants.

New method to produce manifolds homeomorphic but not diffeomorphic to known spaces.

Abstract

One approach to produce a pair of homeomorphic-but-not-diffeomophic closed 4-manifolds is to find a knot which is smoothly slice in one but not the other. This approach has never been run successfully. We give the first examples of a pair of closed 4-manifolds with the same integer cohomology ring where the diffeomorphism type is distinguished by this approach. Along the way, we produce the first examples of 4-manifolds with nonvanishing Seiberg-Witten invariants and the same integer cohomology as $\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$ which are not diffeomorphic to $\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$. We also give a simple new construction of a 4-manifold which is homeomorphic-but-not-diffeomorphic to $\mathbb{C}P^2\#5\overline{\mathbb{C}P^2}$.