scholarly journals Dynamic asymptotic dimension for actions of virtually cyclic groups

2021 ◽  
pp. 1-9
Author(s):  
Massoud Amini ◽  
Kang Li ◽  
Damian Sawicki ◽  
Ali Shakibazadeh
Keyword(s):  
Cyclic Group ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Direct Consequence ◽  
Asymptotic Dimension ◽  
Hausdorff Spaces ◽  
Cyclic Groups ◽  
Finite Dimensional ◽  
Famous Result ◽  
Free Actions

We show that the dynamic asymptotic dimension of an action of an infinite virtually cyclic group on a compact Hausdorff space is always one if the action has the marker property. This in particular covers a well-known result of Guentner, Willett, and Yu for minimal free actions of infinite cyclic groups. As a direct consequence, we substantially extend a famous result by Toms and Winter on the nuclear dimension of $C^{*}$ -algebras arising from minimal free $\mathbb {Z}$ -actions. Moreover, we also prove the marker property for all free actions of countable groups on finite-dimensional compact Hausdorff spaces, generalizing a result of Szabó in the metrisable setting.

Uniqueness of Free Actions on S3 Respecting a Knot

1987 ◽  
Vol 39 (4) ◽  
pp. 969-982 ◽  
Author(s):  
Michel Boileau ◽  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Cyclic Group ◽  
Group Action ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Cyclic Group Action ◽  
Periodic Diffeomorphisms ◽  
Free Actions

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

scholarly journals On projective lift and orbit spaces

1994 ◽  
Vol 50 (3) ◽  
pp. 445-449 ◽  
Author(s):  
T.K. Das
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Covariant Functor ◽  
Hausdorff Spaces ◽  
Orbit Spaces ◽  
Locally Compact ◽  
Discontinuous Group ◽  
Coreflective Subcategory ◽  
Group Of Homeomorphisms ◽  
Locally Compact Hausdorff Space

By constructing the projective lift of a dp-epimorphism, we find a covariant functor E from the category Cd of regular Hausdorff spaces and continuous dp-epimorphisms to its coreflective subcategory εd consisting of projective objects of Cd We use E to show that E(X/G) is homeomorphic to EX/G whenever G is a properly discontinuous group of homeomorphisms of a locally compact Hausdorff space X and X/G is an object of Cd.

scholarly journals On dimension andweight of a local contact algebra

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5481-5500
Author(s):  
G. Dimov ◽  
E. Ivanova-Dimova ◽  
I. Düntsch
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Local Contact ◽  
Continuous Maps ◽  
Contact Algebra ◽  
Contact Algebras ◽  
Locally Compact Hausdorff Space

As proved in [16], there exists a duality ?t between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight wa and of dimension dima of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then w(X) = wa(?t(X)), and if, in addition, X is normal, then dim(X) = dima(?t(X)).

A Banach–Stone theorem for spaces of weak* continuous functions

1985 ◽  
Vol 101 (3-4) ◽  
pp. 203-206 ◽  
Author(s):  
Michael Cambern
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Hausdorff Spaces ◽  
Space Of Continuous Functions ◽  
Dual Banach Space ◽  
Stone Theorem

SynopsisIf X is a compact Hausdorff space and E a dual Banach space, let C(X, Eσ*) denote the Banach space of continuous functions F from X to E when the latter space is provided with its weak * topology, normed by . It is shown that if X and Y are extremally disconnected compact Hausdorff spaces and E is a uniformly convex Banach space, then the existence of an isometry between C(X, Eσ*) and C(Y, Eσ*) implies that X and Y are homeomorphic.

scholarly journals Proximity inductive dimension and Brouwer dimension agree on compact Hausdorff spaces

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1431-1437
Author(s):  
Jeremy Siegert
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Covering Dimension ◽  
Polish Spaces ◽  
Inductive Dimension ◽  
Isolated Points

We show that the proximity inductive dimension defined by Isbell agrees with the Brouwer dimension originally described by Brouwer (for Polish spaces without isolated points) on the class of compact Hausdorff spaces. This shows that Fedorchuk?s example of a compact Hausdorff space whose Brouwer dimension exceeds its Lebesgue covering dimension is an example of a space whose proximity inductive dimension exceeds its proximity dimension as defined by Smirnov. This answers Isbell?s question of whether or not proximity inductive dimension and proximity dimension coincide.

scholarly journals On vector lattice-valued measures II

1986 ◽  
Vol 40 (2) ◽  
pp. 234-252 ◽  
Author(s):  
T. V. Panchapagesan ◽  
Shivappa Veerappa Palled
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Vector Lattice ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Baire Measure ◽  
Strongly Regular ◽  
Locally Compact Hausdorff Space

AbstractFor a weakly (, )-distributive vector lattice V, it is proved that a V {}-valued Baire measure 0 on a locally compact Hausdorff space T admits uniquely regular Borel and weakly Borel extensions on T if and only if 0 is strongly regular at . Consequently, for such a vector lattice V every V-valued Baire measure on a locally compact Hausdorff space T has unique regular Borel and weakly Borel extensions. Finally some characterisations of a weakly (, )-distributive vector lattice are given in terms of the existence of regular Borel (weakly Borel) extensions of certain V {}-valued Barie measures on locally compact Hausdorff spaces.

scholarly journals Minimal Hausdorff and maximal compact spaces

1963 ◽  
Vol 3 (2) ◽  
pp. 167-171 ◽  
Author(s):  
N. Smythe ◽  
C. A. Wilkins
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Hausdorff Topology ◽  
Compact Spaces ◽  
Compact Topology ◽  
Non Hausdorff Spaces

Given two topologies J1, J2 on a set X, J1 is said to be coarser than J2, written J1 ≦ J2, if every set open under J1 is open under J2. A minimal Hausdorff space is then one for which there is no coarser Hausdorff topology etc. Vaidyanathaswamy [4] showed that every compact Hausdorff space is both maximal compact and minimal Hausdorff. This raised the question of whether there exist minimal Hausdorff non-compact spaces and/or maximal compact non-Hausdorff spaces. These questions were in fact answered in the affirmative by Ramanathan [2], Balachandran [1], and Hing Tong [3]. Their examples were, however, all on countable sets, and the topology constructed to answer one question bore no relation to the topology answering the second. In particular, the minimal Hausdorff non-compact topologies were not finer than any maximal compact topology.

scholarly journals A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽  
2021 ◽  
Author(s):  
Péter Vrana
Keyword(s):  
Compact Hausdorff Space ◽  
Polynomial Growth ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Equivalent Condition ◽  
Real Numbers ◽  
Asymptotic Spectrum ◽  
Ring Of Continuous Functions ◽  
Archimedean Property ◽  
Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

On the continuity of lattice isomorphisms on C(X, I)

2021 ◽  
Vol 71 (6) ◽  
pp. 1477-1486
Author(s):  
Vahid Ehsani ◽  
Fereshteh Sady
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Unit Interval ◽  
Lattice Isomorphism

Abstract We investigate topological conditions on a compact Hausdorff space Y, such that any lattice isomorphism φ : C(X, I) → C(Y, I), where X is a compact Hausdorff space and I is the unit interval [0, 1], is continuous. It is shown that in either of cases that the set of G δ points of Y has a dense pseudocompact subset or Y does not contain the Stone-Čech compactification of ℕ, such a lattice isomorphism is a homeomorphism.

scholarly journals Complete Bredon cohomology and its applications to hierarchically defined groups

2016 ◽  
Vol 161 (1) ◽  
pp. 143-156
Author(s):  
BRITA E. A. NUCINKIS ◽  
NANSEN PETROSYAN
Keyword(s):  
Integral Characteristic ◽  
Linear Groups ◽  
Dimensional Model ◽  
Classifying Space ◽  
Soluble Groups ◽  
3 Dimensional ◽  
Cyclic Groups ◽  
Bredon Cohomology ◽  
Finite Dimensional ◽  
The Family

AbstractBy considering the Bredon analogue of complete cohomology of a group, we show that every group in the class$\cll\clh^{\mathfrak F}{\mathfrak F}$of type Bredon-FP∞admits a finite dimensional model for$E_{\frak F}G$.We also show that abelian-by-infinite cyclic groups admit a 3-dimensional model for the classifying space for the family of virtually nilpotent subgroups. This allows us to prove that for$\mathfrak {F}$, the class of virtually cyclic groups, the class of$\cll\clh^{\mathfrak F}{\mathfrak F}$-groups contains all locally virtually soluble groups and all linear groups over${\mathbb{C}}$of integral characteristic.

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