Applied Mathematics Commons^™

Relationships Between Topological Properties Of X And Algebraic Properties Of Intermediate Rings A(X), Joshua Sack

Summer Conference on Topology and Its Applications

A topological property is a property invariant under homeomorphism, and an algebraic property of a ring is a property invariant under ring isomorphism. Let C(X) be the ring of real-valued continuous functions on a Tychonoff space X, let C^*(X) ⊆ C(X) be the subring of those functions that are bounded, and call a ring A(X) an intermediate ring if C^*(X) ⊆ A(X) ⊆ C(X). For a class Q of intermediate rings, an algebraic property P describes a topological property T among Q if for all A(X), B(Y) ∈ Q if A(X) and B(Y) both satisfy P, …

Fiber Strong Shape Theory For Topological Spaces, Ruslan Tsinaridze, Vladimer Baladze

Summer Conference on Topology and Its Applications

The purpose of this paper is the construction and investigation of fiber strong shape theory for compact metrizable spaces over a fixed base space B₀ , using the fiber versions of cotelescop, fibrant space and SSDR-map. In the paper obtained results containing the characterizations of fiber strong shape equivalences, based on the notion of double mapping cylinder over a fixed space B₀. Besides, in the paper we construct and develop a fiber strong shape theory for arbitrary spaces over fixed metrizable space B₀. Our approach is based on the method of Mardešić-Lisica and instead of …

Highly Hamiltonian Graphs And Digraphs, Zhenming Bi

Dissertations

A cycle that contains every vertex of a graph or digraph is a Hamiltonian cycle. A graph or digraph containing such a cycle is itself called Hamiltonian. This concept is named for the famous Irish physicist and mathematician Sir William Rowan Hamilton. These graphs and digraphs have been the subject of study for over six decades. In this dissertation, we study graphs and digraphs with even stronger Hamiltonian properties, namely highly Hamiltonian graphs and digraphs.

On The Chogoshvili Homology Theory Of Continuous Maps Of Compact Spaces, Anzor Beridze, Vladimer Baladze

Summer Conference on Topology and Its Applications

In this paper an exact homology functor from the category Mor_C of continuous maps of compact Hausdorff spaces to the category LES of long exact sequences of abelian groups is defined (cf. [2], [3], [5]). This functor is an extension of the Hu homology theory, which is uniquely defined on the category of continuous maps of finite CW complexes and is constructed without the relative homology groups [9]. To define the given homology functor we use the Chogoshvili construction of projective homology theory [7], [8]. For each continuous map f:X → Y of compact spaces, using the notion of …

Spaces With No S Or L Subspaces, Joan Hart, Kenneth Kunen

Summer Conference on Topology and Its Applications

We show it consistent for spaces X and Y to be both HS and HL even though their product X ×Y contains an S-space. Recall that an S-space is a T₃ space that is HS but not HL.

More generally, consider spaces that contain neither an S-space nor an L-space. We say a space is ESLC iff each of its subspaces is either both HS and HL or neither HS nor HL. The "C" in "ESLC" refers to HC; a space is HC iff each of its subspaces has the ccc (countable chain condition) (iff the space has no …

The Specification Property And Infinite Entropy For Certain Classes Of Linear Operators, James Kelly, Will Brian, Tim Tennant

Summer Conference on Topology and Its Applications

We study the specification property and infinite topological entropy for two specific types of linear operators: translation operators on weighted Lebesgue function spaces and weighted backward shift operators on sequence F-spaces.

It is known from the work of Bartoll, Martinínez-Giménez, Murillo-Arcila (2014), and Peris, that for weighted backward shift operators, the existence of a single non-trivial periodic point is sufficient for specification. We show this also holds for translation operators on weighted Lebesgue function spaces. This implies, in particular, that for these operators, the specification property is equivalent to Devaney chaos. We also show that these forms of chaos imply …

Some Applications Of The Point-Open Subbase Game, David Guerrero Sanchez

Summer Conference on Topology and Its Applications

Given a subbase S of a space X, the game PO(S,X) is defined for two players P and O who respectively pick, at the n-th move, a point xn 2 X and a set Un 2 S such that xn 2 Un . The game stops after the moves {xn, Un : n 2 !} have been made and the player P wins if the union of the Un’s equals X; otherwise O is the winner. Since PO(S,X) is an evident modification of the well-known point-open game PO(X), the primary line of research is to describe the relationship between PO(X) …

Cohen Reals And The Sequential Order Of Groups, Alexander Shibakov

Summer Conference on Topology and Its Applications

We show that adding uncountably many Cohen reals to a model of diamond results in a model with no countable sequential group with an intermediate sequential order. The same model has an uncountable group of sequential order 2. We also discuss related questions.

Entropy Of Induced Continuum Dendrite Homeomorphisms, Jennyffer Bohorquez, Alexander Arbieto

Summer Conference on Topology and Its Applications

Let f: D → D be a dendrite homeomorphism. Let C(D) denote the hyperspace of all nonempty connected compact subsets of D endowed with the Hausdorff metric. Let C(f):C(D) → C(D) be the induced continuum homeomorphism. In this talk we sketch the proof of the following result: If there exists a nonrecurrent branch point then the topological entropy of C(f) is ∞.

Relationships Between Hereditary Sobriety, Sobriety, Td, T1, And Locally Hausdorff, Stephen Rodabaugh, Jeffrey T. Denniston, Austin Melton, Jamal K. Tartir

Summer Conference on Topology and Its Applications

This work augments the standard relationships between sobriety, T₁, and Hausdorff by mixing in locally Hausdorff and the compound axioms sober + T₁ and sober + T_D. We show the latter compound condition characterizes hereditary sobriety, and that locally Hausdorff fits strictly between Hausdorff and sober + T₁. Classes of examples are constructed, in part to show the non-reversibility of key implications.

Some New Completeness Properties In Topological Spaces, Cetin Vural, Süleyman Önal

Summer Conference on Topology and Its Applications

One of the most widely known completeness property is the completeness of metric spaces and the other one being of a topological space in the sense of Cech. It is well known that a metrizable space X is completely metrizable if and only if X is Cech-complete. One of the generalisations of completeness of metric spaces is subcompactness. It has been established that, for metrizable spaces, subcompactness is equivalent to Cech-completeness. Also the concept of domain representability can be considered as a completeness property. In [1], Bennett and Lutzer proved that Cech-complete spaces are domain representable. They also proved, in …

Shift Maps And Their Variants On Inverse Limits With Set-Valued Functions, Judy Kennedy, Kazuhiro Kawamura, Van Nall, Goran Erceg

Summer Conference on Topology and Its Applications

We study inverse limits with set-valued functions using a pull-back construction and representing the space as an ordinary inverse limit space, which allows us to prove some known results and their extensions in a unified scheme. We also present a scheme to construct shift dynamics on the limit space and give some examples using the construction.

Sequential Decreasing Strong Size Properties, Miguel A. Lara, Fernando Orozco, Felix Capulín

Summer Conference on Topology and Its Applications

Let X be a continuum. A topological property P is said to be a sequential decreasing strong size property provided that if μ is a strong size map for C_n(X), {t_n} is a sequence in the interval (t, 1) such that limt_n = t and each fiber μ^-1 (t_n) has the property P, then μ^-1 (t) has the property P. We show that the following properties are sequential decreasing strong size properties: be a Kelley continuum, indecomposability, local connectedness, continuum chainability and unicoherence.

On A Construction Of Some Class Of Metric Spaces, Dariusz Bugajewski

Summer Conference on Topology and Its Applications

In this talk we are going to describe Sz´az’s construction of some class of metric spaces. Most of all we will analyze topological properties of metric spaces obtained by using Sz´az’s construction. In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we will discuss the relation between Sz´az’s construction and the “linking construction”. A particular attention will be drawn to the “floor” metric, the analysis of which provides some interesting observations.

Lifting Homeomorphisms Of Cyclic Branched Covers Of The Sphere, Rebecca R. Winarski, Tyrone Ghaswala

Summer Conference on Topology and Its Applications

Birman and Hilden ask: given finite branched cover X over the 2-sphere, does every homeomorphism of the sphere lift to a homeomorphism of X? For covers of degree 2, the answer is yes, but the answer is sometimes yes and sometimes no for higher degree covers. In joint work with Ghaswala, we completely answer the question for cyclic branched covers. When the answer is yes, there is an embedding of the mapping class group of the sphere into a finite quotient of the mapping class group of X. In a family where the answer is no, we find a presentation …

Totally Geodesic Surfaces In Arithmetic Hyperbolic 3-Manifolds, Benjamin Linowitz, Jeffrey S. Meyer

Summer Conference on Topology and Its Applications

In this talk we will discuss some recent work on the problem of determining the extent to which the geometry of an arithmetic hyperbolic 3-manifold M is determined by the geometric genus spectrum of M (i.e., the set of isometry classes of finite area, properly immersed, totally geodesic surfaces of M, considered up to free homotopy). In particular, we will give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum and analyze the growth of the genera of minimal surfaces across commensurability classes. These results have applications to the study of …

Aperiodic Colorings And Dynamics, Ramon Barral Lijo, Jesús A. Álvarez López

Summer Conference on Topology and Its Applications

A graph coloring is strongly aperiodic if every colored graph in its hull has no automorphisms. The talk will describe a method to define strongly aperiodic colorings on graphs with bounded degree. This also provides an optimal bound for the strongly distinguishing number of a graph. Then some applications to the theory of foliated spaces and to tilings will be discussed.

Classifying Matchbox Manifolds, Olga Lukina

Summer Conference on Topology and Its Applications

A matchbox manifold is a compact connected foliated space, locally homeomorphic to the product of a Euclidean disk and a Cantor set. Strange attractors in dynamical systems, and exceptional minimal sets of smooth foliations present examples of matchbox manifolds. Many actions of profinite groups on trees can be suspended to obtain matchbox manifolds, and similar examples arise in other contexts and in other parts of mathematics.

Thus there is a natural problem of classifying matchbox manifolds. The most tractable class of matchbox manifolds is the class of weak solenoids which are the inverse limits of finite-to-one coverings of closed manifolds. …

Quotients Of N-Fold Hyperspaces, Sergio Macías, Javier Camargo

Summer Conference on Topology and Its Applications

iven a continuum X and an integer n ≥ 2, let C_n(X) be the n-fold hyperspace of X consisting of all nonempty closed subsets of X with at most n components. We consider the quotient space Cⁿ₁(X)=C_n(X)/C₁(X) with the quotient topology. We prove several properties. For example: Cⁿ₁(X) is unicoherent; if X has the property of Kelley, Cⁿ₁(X) is contractible; dim(C_n(X))=dim(Cⁿ₁(X)); both Cⁿ₁([0, 1]) and Cⁿ₁(S¹) are Cantor manifolds; etc.

On Cohomological Dimensions Of Remainders Of Stone-Čech Compactifications, Vladimer Baladze

Summer Conference on Topology and Its Applications

In the paper the necessary and sufficient conditions are found under which a metrizable space has the Stone-Cech compactification whose remainder has the given cohomological dimensions (cf. [Sm], Problem I, p.332 and Problem II, p.334, and [A-N]).

In the paper [B] an outline of a generalization of Cech homology theory was given by replacing the set of all finite open coverings in the definition of Cech (co)homology group (Ĥⁿ_f(X, A;G)) Ĥ_n^f(X, A;G) (see [E-S], Ch.IX, p.237) by the set of all finite open families of border open coverings [Sm₁].

Following Y. Kodama …