scholarly journals Eilenberg–Mac Lane Spaces for Topological Groups

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 90
Author(s):  
Ged Corob Cook
Keyword(s):  
Careful Analysis ◽  
Profinite Groups ◽  
Topological Groups ◽  
Hausdorff Spaces ◽  
Topological Version ◽  
Group Object ◽  
Cartesian Closed ◽  
Model Structures ◽  
Mac Lane ◽  
Compactly Generated

In this paper, we establish a topological version of the notion of an Eilenberg–Mac Lane space. If X is a pointed topological space, π 1 ( X ) has a natural topology coming from the compact-open topology on the space of maps S 1 → X . In general, the construction does not produce a topological group because it is possible to create examples where the group multiplication π 1 ( X ) × π 1 ( X ) → π 1 ( X ) is discontinuous. This discontinuity has been noticed by others, for example Fabel. However, if we work in the category of compactly generated, weakly Hausdorff spaces, we may retopologise both the space of maps S 1 → X and the product π 1 ( X ) × π 1 ( X ) with compactly generated topologies to see that π 1 ( X ) is a group object in this category. Such group objects are known as k-groups. Next we construct the Eilenberg–Mac Lane space K ( G , 1 ) for any totally path-disconnected k-group G. The main point of this paper is to show that, for such a G, π 1 ( K ( G , 1 ) ) is isomorphic to G in the category of k-groups. All totally disconnected locally compact groups are k-groups and so our results apply in particular to profinite groups, answering a question of Sauer’s. We also show that analogues of the Mayer–Vietoris sequence and Seifert–van Kampen theorem hold in this context. The theory requires a careful analysis using model structures and other homotopical structures on cartesian closed categories as we shall see that no theory can be comfortably developed in the classical world.

scholarly journals Comparing Cartesian closed categories of (core) compactly generated spaces

2004 ◽  
Vol 143 (1-3) ◽  
pp. 105-145 ◽  
Author(s):  
Martín Escardó ◽  
Jimmie Lawson ◽  
Alex Simpson
Keyword(s):  
Cartesian Closed Categories ◽  
Cartesian Closed ◽  
Compactly Generated

scholarly journals LOCALLY COMPACT WREATH PRODUCTS

2018 ◽  
Vol 107 (1) ◽  
pp. 26-52 ◽  
Author(s):  
YVES CORNULIER
Keyword(s):  
Normal Subgroup ◽  
Wreath Product ◽  
Natural Extension ◽  
Wreath Products ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Intermediate Growth ◽  
Compactly Generated

Wreath products of nondiscrete locally compact groups are usually not locally compact groups, nor even topological groups. As a substitute introduce a natural extension of the wreath product construction to the setting of locally compact groups. Applying this construction, we disprove a conjecture of Trofimov, constructing compactly generated locally compact groups of intermediate growth without any open compact normal subgroup.

Reflexive objects in topological categories

1996 ◽  
Vol 6 (4) ◽  
pp. 375-386
Author(s):  
Michael D. Rice
Keyword(s):  
Geometric Structure ◽  
Unit Interval ◽  
Continuous Image ◽  
Topological Category ◽  
Hausdorff Spaces ◽  
Tychonoff Space ◽  
Countably Compact ◽  
Discrete Subspace ◽  
Cartesian Closed

This paper presents several basic results about the non-existence of reflexive objects in cartesian closed topological categories of Hausdorff spaces. In particular, we prove that there are no non-trivial countably compact reflexive objects in the category of Hausdorff k-spaces and, more generally, that any non-trivial reflexive Tychonoff space in this category contains a closed discrete subspace corresponding to a numeral system in the sense of Wadsworth. In addition, we establish that a reflexive Tychonoff space in a cartesian-closed topological category cannot contain a non-trivial continuous image of the unit interval. Therefore, if there exists a non-trivial reflexive Tychonoff space, it does not have a nice geometric structure.

scholarly journals UNIVERSAL SUBGROUPS OF POLISH GROUPS

2014 ◽  
Vol 79 (4) ◽  
pp. 1148-1183 ◽  
Author(s):  
KONSTANTINOS A. BEROS
Keyword(s):  
Topological Group ◽  
Banach Spaces ◽  
Topological Groups ◽  
Locally Compact ◽  
Polish Groups ◽  
Countable Power ◽  
Compact Case ◽  
Image Position ◽  
The Relationship ◽  
Compactly Generated

AbstractGiven a class${\cal C}$of subgroups of a topological groupG, we say that a subgroup$H \in {\cal C}$is auniversal${\cal C}$subgroupofGif every subgroup$K \in {\cal C}$is a continuous homomorphic preimage ofH. Such subgroups may be regarded as complete members of${\cal C}$with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable powerGωhas a universalKσsubgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universalKσand compactly generated subgroups. As an aside, we explore the relationship between the classes ofKσand compactly generated subgroups and give conditions under which the two coincide.

scholarly journals On Følner sets in topological groups

2018 ◽  
Vol 154 (7) ◽  
pp. 1333-1361 ◽  
Author(s):  
Friedrich Martin Schneider ◽  
Andreas Thom
Keyword(s):  
Topological Group ◽  
Neumann Problem ◽  
Affirmative Answer ◽  
Left Translation ◽  
Topological Groups ◽  
Von Neumann ◽  
Topological Version ◽  
Følner Sets ◽  
Uniform Manner

We extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set $G$. As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.

LOCALLY COMPACT GROUPS ADMITTING FAITHFUL STRONGLY CHAOTIC ACTIONS ON HAUSDORFF SPACES

2013 ◽  
Vol 23 (09) ◽  
pp. 1350158 ◽  
Author(s):  
FRIEDRICH MARTIN SCHNEIDER ◽  
SEBASTIAN KERKHOFF ◽  
MIKE BEHRISCH ◽  
STEFAN SIEGMUND
Keyword(s):  
Hausdorff Space ◽  
Geometric Characterization ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Continuous Action

In this paper we provide a geometric characterization of those locally compact Hausdorff topological groups which admit a faithful strongly chaotic continuous action on some Hausdorff space.

scholarly journals Rational discrete first degree cohomology for totally disconnected locally compact groups

2018 ◽  
Vol 168 (2) ◽  
pp. 361-377 ◽  
Author(s):  
ILARIA CASTELLANO
Keyword(s):  
Cohomological Dimension ◽  
Finite Graph ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Fundamental Groups ◽  
Profinite Groups ◽  
Locally Compact ◽  
Number Of Ends ◽  
Totally Disconnected ◽  
Compactly Generated

AbstractIt is well known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group G can be detected on the cohomology group H1(G,R[G]), where R is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group G the same information about the number of ends of G in the sense of H. Abels can be provided by dH1(G, Bi(G)), where Bi(G) is the rational discrete standard bimodule of G, and dH•(G, _) denotes rational discrete cohomology as introduced in [6].As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B).

scholarly journals Topological groups with dense compactly generated subgroups

2002 ◽  
Vol 3 (1) ◽  
pp. 85 ◽  
Author(s):  
Hiroshi Fujita ◽  
Dimitri Shakhmatov
Keyword(s):  
Topological Group ◽  
Compact Subset ◽  
Open Subset ◽  
Identity Element ◽  
Topological Groups ◽  
Finitely Generated ◽  
Finite Set ◽  
Metrizable Topological Group ◽  
Compactly Generated ◽  
Compact Subsets

<p>A topological group G is: (i) compactly generated if it contains a compact subset algebraically generating G, (ii) -compact if G is a union of countably many compact subsets, (iii) <sub>0</sub>-bounded if arbitrary neighborhood U of the identity element of G has countably many translates xU that cover G, and (iv) finitely generated modulo open sets if for every non-empty open subset U of G there exists a finite set F such that F  U algebraically generates G. We prove that: (1) a topological group containing a dense compactly generated subgroup is both <sub>0</sub>-bounded and finitely generated modulo open sets, (2) an almost metrizable topological group has a dense compactly generated subgroup if and only if it is both <sub>0</sub>-bounded and finitely generated modulo open sets, and (3) an almost metrizable topological group is compactly generated if and only if it is -compact and finitely generated modulo open sets.</p>

scholarly journals Algebras with actions and automata

1982 ◽  
Vol 5 (1) ◽  
pp. 61-85
Author(s):  
W. Kühnel ◽  
M. Pfender ◽  
J. Meseguer ◽  
I. Sols
Keyword(s):  
Axiomatic Approach ◽  
Structure Theory ◽  
Topological Spaces ◽  
Algebraic Structures ◽  
Hausdorff Spaces ◽  
Left Action ◽  
Common Structure ◽  
The One ◽  
Cartesian Closed ◽  
In The Beginning

In the present paper we want to give a common structure theory of left action, group operations,R-modules and automata of different types defined over various kinds of carrier objects: sets, graphs, presheaves, sheaves, topological spaces (in particular: compactly generated Hausdorff spaces). The first section gives an axiomatic approach to algebraic structures relative to a base categoryB, slightly more powerful than that of monadic (tripleable) functors. In section2we generalize Lawveres functorial semantics to many-sorted algebras over cartesian closed categories. In section3we treat the structures mentioned in the beginning as many-sorted algebras with fixed “scalar” or “input” object and show that they still have an algebraic (or monadic) forgetful functor (theorem 3.3) and hence the general theory of algebraic structures applies. These structures were usually treated as one-sorted in the Lawvere-setting, the action being expressed by a family of unary operations indexed over the scalars. But this approach cannot, as the one developed here, describe continuity of the action (more general: the action to be aB-morphism), which is essential for the structures mentioned above, e.g. modules for a sheaf of rings or topological automata. Finally we discuss consequences of theorem 3.3 for the structure theory of various types of automata. The particular case of algebras with fixed “natural numbers object” has been studied by the authors in [23].

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