Minimal genera of open 4-manifolds

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.Comment: 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 detail

Locally holomorphic maps yield symplectic structures

For a smooth map $f:X^4\to\Sigma^2$ that is locally modeled by holomorphic maps, the domain is shown to admit a symplectic structure that is symplectic on some regular fiber, if and only if $f^*[\Sigma]\ne0$. If so, the space of symplectic forms on $X$ that are symplectic on all fibers is nonempty and contractible. The cohomology classes of these forms vary with the maximum possible freedom on the reducible fibers, subject to the obvious constraints. The above results are derived via an analogous theorem for locally holomorphic maps $f:X^{2n}\to Y^{2n-2}$ with $Y$ symplectic.Comment: 10 pages, no figure