Brendon LaBuz - Academia.edu (original) (raw)

Papers by Brendon LaBuz

Topology and its Applications, Aug 1, 2011

In "Rips complexes and covers in the uniform category" [4] the authors define, following James [5... more In "Rips complexes and covers in the uniform category" [4] the authors define, following James [5], covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper extends these results by investigating the existence of these covering maps relative to subgroups of the uniform fundamental group and the fundamental group of the base space.

arXiv (Cornell University), Apr 5, 2011

It is known that shape injectivity implies homotopical Hausdorff and that the converse does not h... more It is known that shape injectivity implies homotopical Hausdorff and that the converse does not hold, even if the space is required to be a Peano continuum. This paper gives an alternative definition of homotopical Hausdorff inspired by a new topology on the set of fixed endpoint homotopy classes of paths. This version is equivalent to shape injectivity for Peano spaces. Contents 1. Introduction 1 2. Shape injectivity 1 3. Homotopical Hausdorff 5 References 7

arXiv (Cornell University), Jan 16, 2009

Berestovskii and Plaut introduced the concept of a coverable uniform space [1] when developing th... more Berestovskii and Plaut introduced the concept of a coverable uniform space [1] when developing their theory of generalized universal covering maps for uniform spaces. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a locally uniformly joinable uniform space [2] when developing their theory of generalized uniform covering maps which was motivated by [1]. It is easy to see that a chain connected coverable uniform space is locally uniformly joinable. This paper points out an error in the attempt in [5] to prove that a locally uniformly joinable chain connected uniform space is coverable.

arXiv (Cornell University), Mar 30, 2014

The set of homotopy classes of based paths in the Hawaiian earring has a natural R-tree structure... more The set of homotopy classes of based paths in the Hawaiian earring has a natural R-tree structure, but under that metric the action by the fundamental group is not by isometries. Motivated by a suggestion by James W. Cannon and Gregory R. Conner, this paper defines an R ω-metric that does admit for an isometric action by the fundamental group. The space does not become an R ω-tree but is 0-hyperbolic and embeds in an R ω-tree. Cannon and Conner define big free groups BF(c) for cardinal number c which are a generalization of the fundamental group of the Hawaiian earring. They define a big Cayley graph which coincides with the set of homotopy classes of paths in the case of the Hawaiian earring. Instead of inserting real intervals to obtain the Cayley graph, we can insert R c-intervals and obtain a new R c-tree which admits an isometric action. In fact we do not need all of R c ; we can insert Z c-intervals and obtain a Z c-tree. In the case of the Hawaiian earring we give a combinatorial description of the Z ω-tree and the corresponding action. Contents 1. Introduction 1 2. Big free groups and Λ-trees 2 3. Big free groups acting on the big Cayley graph 8 4. A combinatorial description 9 5. The induced topology 11 References 11 2010 Mathematics Subject Classification. 20E08, 20F67.

arXiv (Cornell University), Jan 12, 2021

Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts of uniform joinability and local unifor... more Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts of uniform joinability and local uniform joinability for uniform spaces when developing their theory of generalized uniform covering maps which was motivated by a paper by Berestovskii and Plaut. (Local) uniform joinability can be thought of as analogous to (local) path connectedness. A chain connected locally uniformly joinable uniform space is uniformly joinable. This note gives an example of a metric space that is uniformly joinable but not locally uniformly joinable. Plaut recently defined the concept of a weakly chained uniform space. We show that a weakly chained metrizable uniform space is locally uniformly joinable. Since local uniform joinability is equivalent to pointed 1-movability for metric continua, we find that weak chainability is equivalent to pointed 1movability for such spaces. Contents 1. Introduction 1 2. The Texas circle 2 3. Weakly chained spaces 3 References 4

arXiv (Cornell University), Dec 7, 2011

This note extends the invariant defined in "An invariant of metric spaces under bornologous equiv... more This note extends the invariant defined in "An invariant of metric spaces under bornologous equivalences" to the coarse category.

arXiv (Cornell University), Dec 7, 2011

In [2] an invariant of metric spaces under bornologous equivalences is defined. In [3] this invar... more In [2] an invariant of metric spaces under bornologous equivalences is defined. In [3] this invariant is extended to coarse equivalences. In both papers the invariant is defined for a class of metric spaces called sigma stable. This paper extends the invariant to all metric spaces and also gives an example of a space that is not sigma stable. Contents 1. Introduction 1 2. The invariant 2 3. A space that is not σ-stable 4 Appendix A. Direct limits 6 References 6

Topology and its Applications, May 1, 2012

Rips complexes and covers in the uniform category" develops a theory of generalized uniform cover... more Rips complexes and covers in the uniform category" develops a theory of generalized uniform covering maps. In particular, if X is uniformly joinable and chain connected, then the endpoint map GP (X, x 0) → X from the space of generalized paths in X starting at x 0 to X is a universal generalized uniform covering map. I.M. James introduced uniform covering maps in his 1990 book "Introduction to Uniform Spaces". This paper notes that GP (X, x 0) → X is uniformly equivalent to the inverse limit of uniform covering maps and is therefore approximated by uniform covering maps. A characterization of generalized uniform covering maps that are approximated by uniform covering maps is provided as well as a characterization of generalized uniform covering maps that are uniformly equivalent to the inverse limit of uniform covering maps. Inverse limits of group actions that induce generalized uniform covering maps are also treated.

Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts of uniform joinability and local unifor... more Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts of uniform joinability and local uniform joinability for uniform spaces when developing their theory of generalized uniform covering maps which was motivated by a paper by Berestovskii and Plaut. (Local) uniform joinability can be thought of as analogous to (local) path connectedness. A chain connected locally uniformly joinable uniform space is uniformly joinable. This note gives an example of a metric space that is uniformly joinable but not locally uniformly joinable. Plaut recently defined the concept of a weakly chained uniform space. We show that a weakly chained metrizable uniform space is locally uniformly joinable. Since local uniform joinability is equivalent to pointed 1-movability for metric continua, we find that weak chainability is equivalent to pointed 1movability for such spaces.

Berestovskii and Plaut introduced the concept of a coverable uniform space when developing their ... more Berestovskii and Plaut introduced the concept of a coverable uniform space when developing their theory of generalized universal covering maps for uniform spaces. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a locally uniformly joinable uniform space when developing their theory of generalized uniform covering maps which was motivated by the work of Berestovskii and Plaut. It is easy to see that a chain connected coverable uniform space is locally uniformly joinable. This paper points out an error in the attempt in Plaut's "An equivalent condition for a uniform space to be coverable" to prove that a locally uniformly joinable chain connected uniform space is coverable.

Topology and its Applications, 2010

Topology and its Applications, 2012

Topology and its Applications, Aug 1, 2011

In "Rips complexes and covers in the uniform category" [4] the authors define, following James [5... more In "Rips complexes and covers in the uniform category" [4] the authors define, following James [5], covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper extends these results by investigating the existence of these covering maps relative to subgroups of the uniform fundamental group and the fundamental group of the base space.

arXiv (Cornell University), Apr 5, 2011

It is known that shape injectivity implies homotopical Hausdorff and that the converse does not h... more It is known that shape injectivity implies homotopical Hausdorff and that the converse does not hold, even if the space is required to be a Peano continuum. This paper gives an alternative definition of homotopical Hausdorff inspired by a new topology on the set of fixed endpoint homotopy classes of paths. This version is equivalent to shape injectivity for Peano spaces. Contents 1. Introduction 1 2. Shape injectivity 1 3. Homotopical Hausdorff 5 References 7

arXiv (Cornell University), Jan 16, 2009

Berestovskii and Plaut introduced the concept of a coverable uniform space [1] when developing th... more Berestovskii and Plaut introduced the concept of a coverable uniform space [1] when developing their theory of generalized universal covering maps for uniform spaces. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a locally uniformly joinable uniform space [2] when developing their theory of generalized uniform covering maps which was motivated by [1]. It is easy to see that a chain connected coverable uniform space is locally uniformly joinable. This paper points out an error in the attempt in [5] to prove that a locally uniformly joinable chain connected uniform space is coverable.

arXiv (Cornell University), Mar 30, 2014

The set of homotopy classes of based paths in the Hawaiian earring has a natural R-tree structure... more The set of homotopy classes of based paths in the Hawaiian earring has a natural R-tree structure, but under that metric the action by the fundamental group is not by isometries. Motivated by a suggestion by James W. Cannon and Gregory R. Conner, this paper defines an R ω-metric that does admit for an isometric action by the fundamental group. The space does not become an R ω-tree but is 0-hyperbolic and embeds in an R ω-tree. Cannon and Conner define big free groups BF(c) for cardinal number c which are a generalization of the fundamental group of the Hawaiian earring. They define a big Cayley graph which coincides with the set of homotopy classes of paths in the case of the Hawaiian earring. Instead of inserting real intervals to obtain the Cayley graph, we can insert R c-intervals and obtain a new R c-tree which admits an isometric action. In fact we do not need all of R c ; we can insert Z c-intervals and obtain a Z c-tree. In the case of the Hawaiian earring we give a combinatorial description of the Z ω-tree and the corresponding action. Contents 1. Introduction 1 2. Big free groups and Λ-trees 2 3. Big free groups acting on the big Cayley graph 8 4. A combinatorial description 9 5. The induced topology 11 References 11 2010 Mathematics Subject Classification. 20E08, 20F67.

arXiv (Cornell University), Jan 12, 2021

Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts of uniform joinability and local unifor... more Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts of uniform joinability and local uniform joinability for uniform spaces when developing their theory of generalized uniform covering maps which was motivated by a paper by Berestovskii and Plaut. (Local) uniform joinability can be thought of as analogous to (local) path connectedness. A chain connected locally uniformly joinable uniform space is uniformly joinable. This note gives an example of a metric space that is uniformly joinable but not locally uniformly joinable. Plaut recently defined the concept of a weakly chained uniform space. We show that a weakly chained metrizable uniform space is locally uniformly joinable. Since local uniform joinability is equivalent to pointed 1-movability for metric continua, we find that weak chainability is equivalent to pointed 1movability for such spaces. Contents 1. Introduction 1 2. The Texas circle 2 3. Weakly chained spaces 3 References 4

arXiv (Cornell University), Dec 7, 2011

This note extends the invariant defined in "An invariant of metric spaces under bornologous equiv... more This note extends the invariant defined in "An invariant of metric spaces under bornologous equivalences" to the coarse category.

arXiv (Cornell University), Dec 7, 2011

In [2] an invariant of metric spaces under bornologous equivalences is defined. In [3] this invar... more In [2] an invariant of metric spaces under bornologous equivalences is defined. In [3] this invariant is extended to coarse equivalences. In both papers the invariant is defined for a class of metric spaces called sigma stable. This paper extends the invariant to all metric spaces and also gives an example of a space that is not sigma stable. Contents 1. Introduction 1 2. The invariant 2 3. A space that is not σ-stable 4 Appendix A. Direct limits 6 References 6

Topology and its Applications, May 1, 2012

Rips complexes and covers in the uniform category" develops a theory of generalized uniform cover... more Rips complexes and covers in the uniform category" develops a theory of generalized uniform covering maps. In particular, if X is uniformly joinable and chain connected, then the endpoint map GP (X, x 0) → X from the space of generalized paths in X starting at x 0 to X is a universal generalized uniform covering map. I.M. James introduced uniform covering maps in his 1990 book "Introduction to Uniform Spaces". This paper notes that GP (X, x 0) → X is uniformly equivalent to the inverse limit of uniform covering maps and is therefore approximated by uniform covering maps. A characterization of generalized uniform covering maps that are approximated by uniform covering maps is provided as well as a characterization of generalized uniform covering maps that are uniformly equivalent to the inverse limit of uniform covering maps. Inverse limits of group actions that induce generalized uniform covering maps are also treated.

Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts of uniform joinability and local unifor... more Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts of uniform joinability and local uniform joinability for uniform spaces when developing their theory of generalized uniform covering maps which was motivated by a paper by Berestovskii and Plaut. (Local) uniform joinability can be thought of as analogous to (local) path connectedness. A chain connected locally uniformly joinable uniform space is uniformly joinable. This note gives an example of a metric space that is uniformly joinable but not locally uniformly joinable. Plaut recently defined the concept of a weakly chained uniform space. We show that a weakly chained metrizable uniform space is locally uniformly joinable. Since local uniform joinability is equivalent to pointed 1-movability for metric continua, we find that weak chainability is equivalent to pointed 1movability for such spaces.

Berestovskii and Plaut introduced the concept of a coverable uniform space when developing their ... more Berestovskii and Plaut introduced the concept of a coverable uniform space when developing their theory of generalized universal covering maps for uniform spaces. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a locally uniformly joinable uniform space when developing their theory of generalized uniform covering maps which was motivated by the work of Berestovskii and Plaut. It is easy to see that a chain connected coverable uniform space is locally uniformly joinable. This paper points out an error in the attempt in Plaut's "An equivalent condition for a uniform space to be coverable" to prove that a locally uniformly joinable chain connected uniform space is coverable.

Topology and its Applications, 2010

Topology and its Applications, 2012