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, 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.

30 pages, 1 figure. v3 is essentially the version published in Geometry and Topology, obtained from v2 by major streamlining for readability. Several new examples added since v2; see last paragraph of introduction for details

References in corpus (8)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Smooth embeddings with Stein surface images
• Stein surfaces as open subsets of C^2
• Minimal genera of open 4-manifolds
• Nuclei and exotic 4-manifolds
• Exotic smoothings via large R^4's in Stein surfaces
• Creating Stein surfaces by topological isotopy

Cited by in corpus (10)

• Minimal genera of open 4-manifolds
• Exotic smoothings via large R^4's in Stein surfaces
• On uniqueness of end sums and 1-handles at infinity
• Connected sum at infinity and 4-manifolds
• Group actions, corks and exotic smoothings of R^4
• Creating Stein surfaces by topological isotopy
• The Minimal Genus Problem
• Nonexistence of twists and surgeries generating exotic 4-manifolds
• Upper bounds for virtual dimensions of Seiberg-Witten moduli spaces
• Topologically trivial proper 2-knots