arXiv subjects

Denise M. Halverson

Publications and source records attributed to Denise M. Halverson.

6 recordsLinked to original sources

arXiv2013

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 $n\geq 4$ determined by general position properties, the piecewise disjoint arc-disk property is satisfied.

math.GT↗

arXiv2013

Decompositions of $\mathbb{R}^n, n \geq 4,$ into convex sets generate codimension 1 manifold factors

We show that if $G$ is an upper semicontinuous decomposition of $\mathbb{R}^n$, $n \geq 4$, into convex sets, then the quotient space $\mathbb{R}^n/G$ is a codimension one manifold factor. In particular, we show that $\mathbb{R}^n/G$ has the disjoint arc-disk property.

math.GT↗

arXiv2012

The Homogeneous Property of the Hilbert Cube

We demonstrate the homogeneity of the Hilbert Cube. In particular, we construct explicit self-homeomorphisms of the Hilbert cube so that given any two points, a homeomorphism moving one to the other may be realized.

math.GT↗

arXiv2012

Survey on the Generalized R. L. Moore Problem

We give an updated extended survey of results related to the celebrated unsolved generalized R. L. Moore problem. In particular, we address the problem of characterizing codimension one manifold factors, i.e. spaces $X$ having the property that $X \times \mathbb{R}$ is a topological manifold. A main part of the paper is devoted to many efficient general position techniques, which have been used to solve special cases of this problem.

math.GT↗

arXiv2009

Detecting codimension one manifold factors with topographical techniques

We prove recognition theorems for codimension one manifold factors of dimension $n \geq 4$. 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 $X \times \mathbb{R}$ is a manifold in the cases when $X$ is a resolvable generalized manifold of finite dimension $n \geq 3$ 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.

math.GT↗

arXiv2009

The Bing-Borsuk and the Busemann Conjectures

We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every $n$-dimensional homogeneous ANR is a topological $n$-manifold, whereas the Busemann Conjecture asserts that every $n$-dimensional $G$-space is a topological $n$-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.

math.GT↗