2,496 research outputs found
Detecting codimension one manifold factors with topographical techniques
We prove recognition theorems for codimension one manifold factors of
dimension . In particular, we formalize topographical methods and
introduce three ribbons properties: the crinkled ribbons property, the twisted
crinkled ribbons property, and the fuzzy ribbons property. We show that is a manifold in the cases when is a resolvable
generalized manifold of finite dimension with either: (1) the
crinkled ribbons property; (2) the twisted crinkled ribbons property and the
disjoint point disk property; or (3) the fuzzy ribbons property
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
Detecting codimension one manifold factors with the piecewise disjoint arc-disk property and related properties
We show that all finite-dimensional resolvable generalized manifolds with the
piecewise disjoint arc-disk property are codimension one manifold factors. We
then show how the piecewise disjoint arc-disk property and other general
position properties that detect codimension one manifold factors are related.
We also note that in every example presently known to the authors of a
codimension one manifold factor of dimension determined by general
position properties, the piecewise disjoint arc-disk property is satisfied
Commuting families in Hecke and Temperley-Lieb algebras
Abstract
We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group . We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.12
Locally -homogeneous Busemann -spaces
We present short proofs of all known topological properties of general
Busemann -spaces (at present no other property is known for dimensions more
than four). We prove that all small metric spheres in locally -homogeneous
Busemann -spaces are homeomorphic and strongly topologically homogeneous.
This is a key result in the context of the classical Busemann conjecture
concerning the characterization of topological manifolds, which asserts that
every -dimensional Busemann -space is a topological -manifold. We also
prove that every Busemann -space which is uniformly locally -homogeneous
on an orbal subset must be finite-dimensional
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