Absolutely exotic compact 4-manifolds

TL;DR

This paper introduces a method to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible ones, by modifying relatively exotic manifolds with homology cobordisms.

Contribution

It provides a novel technique to produce absolutely exotic structures on 4-manifolds, expanding the understanding of exotic smooth structures beyond relative cases.

Findings

Constructed absolutely exotic structures on compact 4-manifolds with boundary.

Extended the concept of exotic structures to absolute cases, independent of boundary parameterization.

Applied the method to corks to produce examples on contractible 4-manifolds.

Abstract

We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent to W that are absolutely exotic copies of each other. In this context, {\em absolute} means that the exotic structure is not relative to a particular parameterization of the boundary. Our examples are constructed by modifying a relatively exotic manifold by adding an invertible homology cobordism along its boundary. Applying this technique to corks (contractible manifolds with a diffeomorphism of the boundary that does not extend to a diffeomorphism of the interior) gives examples of absolutely exotic smooth structures on contractible 4-manifolds.