Minimal genera of open $4$–manifolds

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 many, 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 ${ℂ}^2$ 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.