31 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
Connected sum at infinity and 4-manifolds
We study connected sum at infinity on smooth, open manifolds. This operation
requires a choice of proper ray in each manifold summand. In favorable
circumstances, the connected sum at infinity operation is independent of ray
choices. For each m at least 3, we construct an infinite family of pairs of
m-manifolds on which the connected sum at infinity operation yields distinct
manifolds for certain ray choices. We use cohomology algebras at infinity to
distinguish these manifolds.Comment: 17 pages, 12 figure
On fundamental groups of quotient spaces
In classical covering space theory, a covering map induces an injection of
fundamental groups. This paper reveals a dual property for certain quotient
maps having connected fibers, with applications to orbit spaces of vector
fields and leaf spaces in general.Comment: 12 pages, 4 figures; added references, keywords, and Remark 1.2;
accepted at Topology and its Application
The end sum of surfaces
End sum is a natural operation for combining two noncompact manifolds and has
been used to construct various manifolds with interesting properties. The
uniqueness of end sum has been well-studied in dimensions three and higher. We
study end sum -- and the more general notion of adding a 1-handle at infinity
-- for surfaces and prove uniqueness results. The result of adding a 1-handle
at infinity to distinct ends of a surface with compact boundary is uniquely
determined by the chosen ends and the orientability of the 1-handle. As a
corollary, the end sum of two surfaces with compact boundary is uniquely
determined by the chosen ends. Unlike uniqueness results in higher dimensions,
which rely on isotopy uniqueness of rays, our results rely fundamentally on a
classification of noncompact surfaces.Comment: 40 pages, 29 figure
Orbit Spaces of Gradient Vector Fields
We study orbit spaces of generalized gradient vector fields for Morse
functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they
are quite structured topologically and are amenable to study. We show that
these orbit spaces are locally contractible. We also show that the quotient map
associated to each such orbit space is a weak homotopy equivalence and has the
path lifting property.Comment: 16 pages, 4 figures; strengthened a main result (Corollary 3.5) and
updated the introduction and the conclusio
Borromean rays and hyperplanes
Three disjoint rays in euclidean 3-space form Borromean rays provided their
union is knotted, but the union of any two components is unknotted. We
construct infinitely many Borromean rays, uncountably many of which are
pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.Comment: 41 pages, 30 figures (19 with captions, 11 inline