On localizing groups of exotic diffeomorphisms of 4-manifolds

TL;DR

This paper investigates when infinite groups of exotic diffeomorphisms of 4-manifolds can be localized to smaller submanifolds, revealing limitations and specific conditions for such localizations.

Contribution

It provides new results on the localization of infinitely generated exotic diffeomorphism groups, including non-localization to certain submanifolds and the absence of universal properties.

Findings

Infinitely generated groups do not localize to rational homology balls.

Many exotic diffeomorphisms are not Dehn twists along homology spheres.

No universal cork or compact 4-manifold for exotic diffeomorphisms.

Abstract

Ruberman in the 90's showed that the group of exotic diffeomorphisms of closed 4-manifolds can be infinitely generated. We provide various results on the question of when such infinite generation can localize to a smaller embedded submanifold of the original manifold. Our results include: (1) All known infinitely generated groups of exotic diffeomorphisms of 4-manifolds detected by families Seiberg-Witten theory do not localize to any topologically (locally-flatly) embedded rational homology balls in the ambient 4-manifold. (2) Many exotic diffeomorphisms cannot be obtained as Dehn twists along homology spheres (under mild assumptions). (3) There is no contractible 4-manifolds with Seifert fibered boundary that have a universal property for exotic diffeomorphisms analogous to a universal cork. In addition, there is no universal compact 4-manifold $W$ such that the set of exotic diffeomorphisms of a 4-manifold can localize to an embedding of $W$. (4) Certain infinite generations of exotic diffeomorphism groups do localize to a non-compact subset $V$ with a small Betti number, but not to any compact subset of $V$. (5) An analogous result holds for mapping class groups of 4-manifolds.