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Homogeneity groups of ends of open 3-manifolds
For every finitely generated abelian group G, we construct an irreducible
open 3-manifold M[subscript G] whose end set is homeomorphic to a Cantor set and
whose homogeneity group is isomorphic to G. The end homogeneity group
is the group of self-homeomorphisms of the end set that extend to homeomorphisms
of the 3-manifold. The techniques involve computing the embedding
homogeneity groups of carefully constructed Antoine-type Cantor sets
made up of rigid pieces. In addition, a generalization of an Antoine Cantor
set using infinite chains is needed to construct an example with integer homogeneity
group. Results about the local genus of points in Cantor sets and
about the geometric index are also used.This is the publisher’s final pdf. The published article is copyrighted by the Mathematical Sciences Publishers and can be found at: http://msp.org/pjm/2014/269-1/index.xhtml.Keywords: Manifold end, Homogeneity group, Defining sequence, Geometric index, Cantor set, Abelian group, Rigidity, Open 3-manifol
The Bing-Borsuk and the Busemann Conjectures
We present two classical conjectures concerning the characterization of
manifolds: the Bing Borsuk Conjecture asserts that every -dimensional
homogeneous ANR is a topological -manifold, whereas the Busemann Conjecture
asserts that every -dimensional -space is a topological -manifold. The
key object in both cases are so-called {\it generalized manifolds}, i.e. ENR
homology manifolds. We look at the history, from the early beginnings to the
present day. We also list several open problems and related conjectures.Comment: We have corrected three small typos on pages 8 and
Peaks in the Hartle-Hawking Wave Function from Sums over Topologies
Recent developments in ``Einstein Dehn filling'' allow the construction of
infinitely many Einstein manifolds that have different topologies but are
geometrically close to each other. Using these results, we show that for many
spatial topologies, the Hartle-Hawking wave function for a spacetime with a
negative cosmological constant develops sharp peaks at certain calculable
geometries. The peaks we find are all centered on spatial metrics of constant
negative curvature, suggesting a new mechanism for obtaining local homogeneity
in quantum cosmology.Comment: 16 pages,LaTeX, no figures; v2: some changes coming from revision of
a math reference: wave function peaks sharp but not infinite; v3: added
paragraph in intro on interpretation of wave functio
Homogeneous matchbox manifolds
We prove that a homogeneous matchbox manifold of any finite dimension is
homeomorphic to a McCord solenoid, thereby proving a strong version of a
conjecture of Fokkink and Oversteegen. The proof uses techniques from the
theory of foliations that involve making important connections between
homogeneity and equicontinuity. The results provide a framework for the study
of equicontinuous minimal sets of foliations that have the structure of a
matchbox manifold.Comment: This is a major revision of the original article. Theorem 1.4 has
been broadened, in that the assumption of no holonomy is no longer required,
only that the holonomy action is equicontinuous. Appendices A and B have been
removed, and the fundamental results from these Appendices are now contained
in the preprint, arXiv:1107.1910v
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