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Articles 1 - 9 of 9
Full-Text Articles in Geometry and Topology
Shrinking Attachment Spaces, Anastasia M. Clements
Shrinking Attachment Spaces, Anastasia M. Clements
West Chester University Graduate Theses, Dissertations, and Final Projects
Gluing constructions such as pushouts and other colimits are often used to attach spaces to- gether in algebraic topology. The weak topology is a natural choice of topology for attachment spaces in the context of CW-complexes and simplicial complexes because of its universal property but is insufficient for gluing together infinitely many spaces and preserving topological properties like compactness or metrizability. In this thesis, we introduce a modification of the weak topology called the shrinking attachment topology, which is defined on a space Y constructed by attaching an infinite sequence of spaces B1, B1, B3 …
Identities For Whitehead Products And Infinite Sums, Jeremy Brazas
Identities For Whitehead Products And Infinite Sums, Jeremy Brazas
Mathematics Faculty Publications
Whitehead products and natural infinite sums are prominent in the higher homotopy groups of the n-dimensional infinite earring space En" role="presentation"> and other locally complicated Peano continua. In this paper, we derive general identities for how these operations interact with each other. As an application, we consider a shrinking wedge of finite (n−1)" role="presentation">-connected CW-complexes and compute the infinite-sum closure W2n−1(X)" role="presentation"> of the set of Whitehead products [α,β]" role="presentation"> in π2n−1(X)" role="presentation"> where α,β∈πn(X)" role="presentation"> are represented in respective sub-wedges that meet only at the basepoint. In particular, we show that W2n−1(X)" role="presentation"> is canonically isomorphic to …
A Natural Pseudometric On Homotopy Groups Of Metric Spaces, Jeremy Brazas, Paul Fabel
A Natural Pseudometric On Homotopy Groups Of Metric Spaces, Jeremy Brazas, Paul Fabel
Mathematics Faculty Publications
For a path-connected metric space (X, d), the n-th homotopy group π n ( X) inherits a natural pseudometric from the n-th iterated loop space with the uniform metric. This pseudometric gives π n ( X) the structure of a topological group and when X is compact, the induced pseudometric topology is independent of the metric d. In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on π n ( X). Our main result is that the pseudometric topology agrees with the shape topology on π n ( X) if X …
Free Quasitopological Groups, Jeremy Brazas, Sarah Emery
Free Quasitopological Groups, Jeremy Brazas, Sarah Emery
Mathematics Faculty Publications
In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group F-q(X) on a space X. We show that free quasitopological groups may be constructed directly as quotient spaces of free semitopological monoids, which are themselves constructed by iterating product spaces equipped with the "cross topology." Using this explicit description of F-q(X), we show that for any T-1 space X, F-q(X) is the direct limit of closed subspaces F-q(X)(n) of words of length at most n. We also prove that the natural map i(n): (sic)(n)(i=0)(X boolean OR X-1)(circle times i) - …
On Maps With Continuous Path Lifting, Jeremy Brazas, Atish Mitra
On Maps With Continuous Path Lifting, Jeremy Brazas, Atish Mitra
Mathematics Faculty Publications
We study a natural generalization of covering projections defined in terms of unique lifting properties. A map p : E -+ X has the continuous path-covering property if all paths in X lift uniquely and continuously (rel. basepoint) with respect to the compactopen topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering …
Elements Of Higher Homotopy Groups Undetectable By Polyhedral Approximation, John K. Aceti, Jeremy Brazas
Elements Of Higher Homotopy Groups Undetectable By Polyhedral Approximation, John K. Aceti, Jeremy Brazas
Mathematics Faculty Publications
When nontrivial local structures are present in a topological space X, a common approach to characterizing the isomorphism type of the n-th homotopy group πn(X, x0) is to consider the image of πn(X, x0) in the nth Cˇ ech homotopy group πˇ n(X, x0) under the canonical homomorphism 9n : πn(X, x0) → πˇ n(X, x0). The subgroup ker(9n) is the obstruction to this tactic as it consists of precisely those elements of πn(X, x0), which cannot be detected by polyhedral approximations to X. In this paper, we use higher dimensional analogues of Spanier groups to characterize ker(9n). In particular, …
Higher Spanier Groups, Johnny Aceti
Higher Spanier Groups, Johnny Aceti
West Chester University Master’s Theses
When non-trivial local structures are present in a topological space X, a common ap- proach to characterizing the isomorphism type of the n-th homotopy group πn(X, x0) is to consider the image of πn(X, x0) in the n-th ˇCech homotopy group ˇπn(X, x0) under the canonical homomorphism Ψn : πn(X, x0) → ˇπn(X, x0). The subgroup ker Ψn is the obstruc- tion to this tactic as it consists of precisely those elements of πn(X, x0), which cannont be detected by polyhedral approximations to X. In this paper we present a definition of higher dimensional analouges of Thick Spanier groups use …
Unique Lifting To A Functor, Mark Myers
Unique Lifting To A Functor, Mark Myers
West Chester University Master’s Theses
We develop a functorial approach to quotient constructions, defining morphisms quotient relative to a functor and the dual concept of unique liftings relative to a functor. Various classes of epimorphism are given detailed analysis and their relationship to quotient morphisms characterized. The behavior of unique lifting morphisms with respect to products, equalizers, and general limits in a category are studied. Applications to generalized covering space theory, coreflective subcategories of topological spaces, topological groups and rings, and Galois theory are explored. Finally, we give conditions for the product of two quotient morphisms to be quotient in a braided monoidal closed category.
Solving Parabolic Interface Problems With A Finite Element Method, Henry Brown
Solving Parabolic Interface Problems With A Finite Element Method, Henry Brown
Mathematics Student Work
Partial differential equations (PDEs) dominate mathematical models given their effectiveness and accuracy at modeling the physical realities which govern the world. Though we have these powerful tools, analytic solutions can only be found in the simplest of cases due to the complexity of PDE models. Thus, efficient and accurate computational methods are needed to approximate solutions to PDE models. One class of these methods are finite element methods which can be used domain to provide close approximations to the PDE model in a finite domain. In this presentation, we discuss the use of a Discontinuous Galerkin (DG) Finite Element Methods …