7,423 research outputs found
On uniqueness of end sums and 1-handles at infinity
For oriented manifolds of dimension at least 4 that are simply connected at
infinity, it is known that end summing is a uniquely defined operation. Calcut
and Haggerty showed that more complicated fundamental group behavior at
infinity can lead to nonuniqueness. The present paper examines how and when
uniqueness fails. Examples are given, in the categories TOP, PL and DIFF, of
nonuniqueness that cannot be detected in a weaker category (including the
homotopy category). In contrast, uniqueness is proved for Mittag-Leffler ends,
and generalized to allow slides and cancellation of (possibly infinite)
collections of 0- and 1-handles at infinity. Various applications are
presented, including an analysis of how the monoid of smooth manifolds
homeomorphic to R^4 acts on the smoothings of any noncompact 4-manifold.Comment: 25 pages, 8 figures. v2: Minor expository improvement
Intersection homology of stratified fibrations and neighborhoods
We derive spectral sequences for the intersection homology of stratified
fibrations and approximate tubular neighborhoods in manifold stratified spaces.
These neighborhoods include regular neighborhoods in PL stratified spaces.Comment: 45 page
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
Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives
Given a geometrically finite hyperbolic cone-manifold, with the cone-singularity sufficiently short, we construct a one-parameter family of cone-manifolds decreasing the cone-angle to zero. We also control the geometry of this one-parameter family via the Schwarzian derivative of the projective boundary and the length of closed geodesics
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