Corks for exotic diffeomorphisms

TL;DR

This paper establishes a localization theorem for exotic diffeomorphisms on 4-manifolds, showing they can be supported on specific submanifolds after stabilization, with implications for understanding exotic smooth structures.

Contribution

It proves a new localization theorem for exotic diffeomorphisms on 4-manifolds, detailing how they can be supported on contractible or wedge-of-spheres submanifolds after stabilization.

Findings

Exotic diffeomorphisms can be supported on contractible submanifolds after stabilization.

Diffeomorphisms requiring more stabilization are supported on submanifolds homotopy equivalent to wedges of 2-spheres.

Implications for known exotic diffeomorphisms are discussed.

Abstract

We prove a localization theorem for exotic diffeomorphisms, showing that every diffeomorphism of a compact simply-connected 4-manifold that is isotopic to the identity after stabilizing with one copy of $S^2 \times S^2$, is smoothly isotopic to a diffeomorphism that is supported on a contractible submanifold. For those that require more than one copy of $S^2 \times S^2$, we prove that the diffeomorphism can be isotoped to one that is supported in a submanifold homotopy equivalent to a wedge of 2-spheres, with null-homotopic inclusion map. We investigate the implications of these results by applying them to known exotic diffeomorphisms.