Minimal genera of open 4-manifolds

TL;DR

This paper explores exotic smoothings of open 4-manifolds, utilizing the minimal genus function and Stein surface techniques to achieve greater control and expand known examples of such manifolds.

Contribution

It introduces new methods using the adjunction inequality for Stein surfaces to construct exotic smoothings with controlled genus functions, broadening the class of known exotic 4-manifolds.

Findings

Every 2-handlebody interior admits an exotic smoothing.

Most such smoothings have infinitely or uncountably many distinguished by genus functions.

Topological submanifolds of complex plane domains can have uncountably many diffeomorphism types with same or controlled genus functions.

Abstract

We study exotic smoothings of open 4-manifolds using the minimal genus function and its analog for end homology. While traditional techniques in open 4-manifold smoothing theory give no control of minimal genera, we make progress by using the adjunction inequality for Stein surfaces. Smoothings can be constructed with much more control of these genus functions than the compact setting seems to allow. As an application, we expand the range of 4-manifolds known to have exotic smoothings (up to diffeomorphism). For example, every 2-handlebody interior (possibly infinite or nonorientable) has an exotic smoothing, and "most" have infinitely, or sometimes uncountably many, distinguished by the genus function and admitting Stein structures when orientable. Manifolds with 3-homology are also accessible. We investigate topological submanifolds of smooth 4-manifolds. Every domain of holomorphy (Stein open subset) in the complex plane $C^2$ is topologically isotopic to uncountably many other diffeomorphism types of domains of holomorphy with the same genus functions, or with varying but controlled genus functions.