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- Abelian groups (2)
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Articles 1 - 26 of 26
Full-Text Articles in Physical Sciences and Mathematics
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 …
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 …
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) - …
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, …
Geometric Realizations Of Abstract Regular Polyhedra With Automorphism Group H3, Mark L. Loyola, Jonn Angel L. Aranas
Geometric Realizations Of Abstract Regular Polyhedra With Automorphism Group H3, Mark L. Loyola, Jonn Angel L. Aranas
Mathematics Faculty Publications
A geometric realization of an abstract polyhedron P is a mapping that sends an i-face to an open set of dimension i. This work adapts a method based on Wythoff construction to generate a full rank realization of an abstract regular polyhedron from its automorphism group Gamma. The method entails finding a real orthogonal representation of Gamma of degree 3 and applying its image to suitably chosen (not necessarily connected) open sets in space. To demonstrate the use of the method, it is applied to the abstract polyhedra whose automorphism groups are isomorphic to the non-crystallographic Coxeter group H3.
K -Isocoronal Tilings, Eduard C. Taganap, Ma. Louise Antonette N. De Las Peñas
K -Isocoronal Tilings, Eduard C. Taganap, Ma. Louise Antonette N. De Las Peñas
Mathematics Faculty Publications
In this article, a framework is presented that allows the systematic derivation of planar edge-to-edge k-isocoronal tilings from tile-s-transitive tilings, s k. A tiling T is k-isocoronal if its vertex coronae form k orbits or k transitivity classes under the action of its symmetry group. The vertex corona of a vertex x of T is used to refer to the tiles that are incident to x. The k-isocoronal tilings include the vertex-k-transitive tilings (k-isogonal) and k-uniform tilings. In a vertex-k- transitive tiling, the vertices form k transitivity classes under its symmetry group. If this tiling consists of regular polygons then …
Primitive Substitution Tilings With Rotational Symmetries, April Lynne D. Say-Awen, Ma. Louise Antonette N. De Las Peñas, Dirk Frettlöh
Primitive Substitution Tilings With Rotational Symmetries, April Lynne D. Say-Awen, Ma. Louise Antonette N. De Las Peñas, Dirk Frettlöh
Mathematics Faculty Publications
This work introduces the idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution. Definitions are given of the substitutions σ6 and σ7 which give rise to aperiodic primitive substitution tilings with dense tile orientations and which are invariant under six- and sevenfold rotations, respectively; the derivation of the symmetry orders of their hulls is also presented.
Classification Of Book Representations Of K6, Dana Rowland
Classification Of Book Representations Of K6, Dana Rowland
Mathematics Faculty Publications
A book representation of a graph is a particular way of embedding a graph in three dimensional space so that the vertices lie on a circle and the edges are chords on disjoint topological disks. We describe a set of operations on book representations that preserves ambient isotopy, and apply these operations to K6, the complete graph with six vertices. We prove there are exactly 59 distinct book representations for K6, and we identify the number and type of knotted and linked cycles in each representation. We show that book representations of K6 contain between one and seven links, and …
Construction Of Weavings In The Plane, Eden Delight Miro, Aliw-Iw Zambrano, Agnes Garciano
Construction Of Weavings In The Plane, Eden Delight Miro, Aliw-Iw Zambrano, Agnes Garciano
Mathematics Faculty Publications
This work develops, in graph-theoretic terms, a methodology for systematically constructing weavings of overlapping nets derived from 2-colorings of the plane. From a 2-coloring, two disjoint simple, connected graphs called nets are constructed. The union of these nets forms an overlapping net, and a weaving map is defined on the intersection points of the overlapping net to form a weaving. Furthermore, a procedure is given for the construction of mixed overlapping nets and for deriving weavings from them.
Coding Strategies, The Choquet Game And Domain Representability, Lynne Yengulalp
Coding Strategies, The Choquet Game And Domain Representability, Lynne Yengulalp
Mathematics Faculty Publications
We prove that if the NONEMPTY player has a winning strategy in the strong Choquet game on a regular space X then NONEMPTY has a winning coding strategy in that game (a strategy that only depends on the previous 2 moves). We also prove that any regular domain representable space is generalized subcompact.
On Color Fixing Groups Associated With Colored Symmetrical Tilings, April Lynne D. Say-Awen, Ma. Louise Antonette N. De Las Peñas, Teofina A. Rapanut
On Color Fixing Groups Associated With Colored Symmetrical Tilings, April Lynne D. Say-Awen, Ma. Louise Antonette N. De Las Peñas, Teofina A. Rapanut
Mathematics Faculty Publications
In this paper, we contribute to the study of colored symmetrical tilings by giving formulas for their associated color fixing groups. In the second part of the paper we provide an application of the results in describing symmetry groups of nanostructures.
Every Scattered Space Is Subcompact, William Fleissner, Vladimir Tkachuk, Lynne Yengulalp
Every Scattered Space Is Subcompact, William Fleissner, Vladimir Tkachuk, Lynne Yengulalp
Mathematics Faculty Publications
We prove that every scattered space is hereditarily subcompact and any finite union of subcompact spaces is subcompact. It is a long-standing open problem whether every Čech-complete space is subcompact. Moreover, it is not even known whether the complement of every countable subset of a compact space is subcompact. We prove that this is the case for linearly ordered compact spaces as well as for ω -monolithic compact spaces. We also establish a general result for Tychonoff products of discrete spaces which implies that dense Gδ-subsets of Cantor cubes are subcompact.
Knots In The Canonical Book Representation Of Complete Graphs, Dana Rowland, Andrea Politano
Knots In The Canonical Book Representation Of Complete Graphs, Dana Rowland, Andrea Politano
Mathematics Faculty Publications
We describe which knots can be obtained as cycles in the canonical book representation of the complete graph Kn, and we conjecture that the canonical book representation of Kn attains the least possible number of knotted cycles for any embedding of Kn. The canonical book representation of Kn contains a Hamiltonian cycle that is a composite knot if and only if n ≥12. When p and q are relatively prime, the (p, q) torus knot is a Hamiltonian cycle in the canonical book representation of K2p+q. …
Coarser Connected Metrizable Topologies, Lynne Yengulalp
Coarser Connected Metrizable Topologies, Lynne Yengulalp
Mathematics Faculty Publications
We show that every metric space, X, with w(⩾) c has a coarser connected metrizable topology.
The Center Of Some Braid Groups And The Farrell Cohomology Of Certain Pure Mapping Class Groups, Craig A. Jensen, Yu Qing Chen, Henry H. Glover
The Center Of Some Braid Groups And The Farrell Cohomology Of Certain Pure Mapping Class Groups, Craig A. Jensen, Yu Qing Chen, Henry H. Glover
Mathematics Faculty Publications
In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the primary part of the Farrell cohomology of a few pure mapping class groups.
The Euler Characteristic Of The Whitehead Automorphism Group Of A Free Product, Craig A. Jensen, Jon Mccammond, John Meier
The Euler Characteristic Of The Whitehead Automorphism Group Of A Free Product, Craig A. Jensen, Jon Mccammond, John Meier
Mathematics Faculty Publications
A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups. In particular, we establish formulae for the Euler characteristics of: the group of Whitehead automorphisms...
The Integral Cohomology Of The Group Of Loops, Craig Jensen, Jon Mccammond, John Meier
The Integral Cohomology Of The Group Of Loops, Craig Jensen, Jon Mccammond, John Meier
Mathematics Faculty Publications
No abstract provided.
The Cohomology Of Right Angled Artin Groups With Group Ring Coefficients, Craig A. Jensen
The Cohomology Of Right Angled Artin Groups With Group Ring Coefficients, Craig A. Jensen
Mathematics Faculty Publications
The cohomology of a right-angled Artin group with group ring coefficients is explicitly presented in terms of the cohomology of its defining flag complex. 2000 Mathematics Subject Classification 20F36 (primary), 57M07 (secondary).
Proper Actions Of Automorphism Groups Of Free Products Of Finite Groups, Craig A. Jensen, Yuqing Chen, Henry H. Glover
Proper Actions Of Automorphism Groups Of Free Products Of Finite Groups, Craig A. Jensen, Yuqing Chen, Henry H. Glover
Mathematics Faculty Publications
If G is a free product of finite groups, let ΣAut1(G) denote all (necessarily symmetric) automorphisms of G that do not permute factors in the free product. We show that a McCullough–Miller and Gutiérrez–Krstić derived (also see Bogley–Krstić) space of pointed trees is an EΣAut1(G)-space for these groups.
Automorphisms Of Free Groups With Boundaries, Craig A. Jensen, Nathalie Wahl
Automorphisms Of Free Groups With Boundaries, Craig A. Jensen, Nathalie Wahl
Mathematics Faculty Publications
The automorphisms of free groups with boundaries form a family of groups An,k closely related to mapping class groups, with the standard automorphisms of free groups as An,0 and (essentially) the symmetric automorphisms of free groups as A0,k. We construct a contractible space Ln,k on which An,k acts with finite stabilizers and finite quotient space and deduce a range for the virtual cohomological dimension of An,k. We also give a presentation of the groups and calculate their first homology group.
Homology Of Holomorphs Of Free Groups, Craig A. Jensen
Homology Of Holomorphs Of Free Groups, Craig A. Jensen
Mathematics Faculty Publications
Let Fn be the free group on n letters and let Aut(Fn), Out(Fn) denote the automorphism group and the outer automorphism group of Fn. In this paper the purpose is to obtain some new results on stability and to calculate the mod-p homology of the holomorph of Fn for odd primes in dimensions 1 and 2, and the rational homology in dimensions from 1 through 5.
A Density Property Of The Tori And Duality, Peter Loth
A Density Property Of The Tori And Duality, Peter Loth
Mathematics Faculty Publications
In this note, a short proof of a recent theorem of D. Dikranjan and M. Tkachenko is given, and their result is extended.
The Possibility Of Impossible Pyramids, Thomas Q. Sibley
The Possibility Of Impossible Pyramids, Thomas Q. Sibley
Mathematics Faculty Publications
No abstract provided.
Rhombic Penrose Tilings Can Be 3-Colored, Thomas Q. Sibley, Stan Wagon
Rhombic Penrose Tilings Can Be 3-Colored, Thomas Q. Sibley, Stan Wagon
Mathematics Faculty Publications
No abstract provided.
Splitting Of The Identity Component In Locally Compact Abelian Groups, Peter Loth
Splitting Of The Identity Component In Locally Compact Abelian Groups, Peter Loth
Mathematics Faculty Publications
In this paper we are concerned with the splitting of the identity component G0 in an LCA group G. As Pontrjagin duality shows, this splitting is to the splitting of the torsion part tA in a discrete abelian group A, if G is assumed to be compact.
Changing Modes Of Thought: Non-Euclidean Geometry And The Liberal Arts, Thomas Q. Sibley
Changing Modes Of Thought: Non-Euclidean Geometry And The Liberal Arts, Thomas Q. Sibley
Mathematics Faculty Publications
No abstract provided.