7,598 research outputs found
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 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
Cubic Partial Cubes from Simplicial Arrangements
We show how to construct a cubic partial cube from any simplicial arrangement
of lines or pseudolines in the projective plane. As a consequence, we find nine
new infinite families of cubic partial cubes as well as many sporadic examples.Comment: 11 pages, 10 figure
- …