scholarly journals Remark on a theorem of E.H. Brown

1973 ◽  
Vol 9 (1) ◽  
pp. 55-60 ◽  
Author(s):  
Michael D. Alder
Keyword(s):  
Homotopy Type ◽  
Full Subcategory ◽  
Topological Spaces ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Homotopy Classes ◽  
Continuous Maps ◽  
Homotopy Functors

The proliferation of classifying spaces in recent years owes much to the theorem of Edgar H. Brown, Jr on the representability of homotopy functors. Since the theorem only gives a representation for functors defined on the category of spaces having the homotopy type of a CW complex, there is some interest in finding conditions under which the domain category may be enlarged. It appears that a version of the theorem holds for any small full subcategory of Htp, the category of topological spaces and homotopy classes of continuous maps, but that the resulting classifying space is generally intractable.

Classifying Algebras for the K-Theory of σ-C*-Algebras

1989 ◽  
Vol 41 (6) ◽  
pp. 1021-1089 ◽  
Author(s):  
N. Christopher Phillips
Keyword(s):  
Hilbert Space ◽  
Topological Space ◽  
Fredholm Operators ◽  
Classifying Space ◽  
Homotopy Classes ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Continuous Maps ◽  
Group Structures

In topology, the representable K-theory of a topological space X is defined by the formulas RK0(X) = [X,Z x BU] and RKl(X) = [X, U], where square brackets denote sets of homotopy classes of continuous maps, is the infinite unitary group, and BU is a classifying space for U. (Note that ZxBU is homotopy equivalent to the space of Fredholm operators on a separable infinite-dimensional Hilbert space.) These sets of homotopy classes are made into abelian groups by using the H-group structures on Z x BU and U. In this paper, we give analogous formulas for the representable K-theory for α-C*-algebras defined in [20].

Suspension of the Lusternik-Schnirelmann Category

1970 ◽  
Vol 22 (6) ◽  
pp. 1129-1132
Author(s):  
William J. Gilbert
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Category Structure ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Cw Complexes ◽  
Lusternik Schnirelmann ◽  
Image Position ◽  
Simply Connected ◽  
Homotopy Invariants

Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],We prove thatandIt is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.

scholarly journals A study of some aspects of topological groups

Filomat ◽  
2007 ◽  
Vol 21 (1) ◽  
pp. 55-65
Author(s):  
M.R. Adhikari ◽  
M. Rahaman
Keyword(s):  
Topological Group ◽  
Group Structure ◽  
Base Point ◽  
Homotopy Category ◽  
Topological Spaces ◽  
Topological Groups ◽  
Homotopy Classes ◽  
Continuous Maps

The aim of this paper is to find a generalization of topological groups. The concept arises out of the investigation to obtain a group structure on the set [X,Y], of homotopy classes of maps from a space X to a given space Y for all X which is natural with respect to X. We also study the generalized topological groups. Finally, associated with each generalized topological group we construct a contra variant functor from the homotopy category of pointed topological spaces and base point preserving continuous maps to the category of groups and homomorphism.

scholarly journals On the homotopy theory of monoids

1989 ◽  
Vol 47 (2) ◽  
pp. 171-185
Author(s):  
Carol M. Hurwitz
Keyword(s):  
Homotopy Type ◽  
Homotopy Theory ◽  
Small Category ◽  
Homotopy Equivalence ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Finitely Presented

AbsractIn this paper, it is shown that any connected, small category can be embedded in a semi-groupoid (a category in which there is at least one isomorphism between any two elements) in such a way that the embedding includes a homotopy equivalence of classifying spaces. This immediately gives a monoid whose classifying space is of the same homotopy type as that of the small category. This construction is essentially algorithmic, and furthermore, yields a finitely presented monoid whenever the small category is finitely presented. Some of these results are generalizations of ideas of McDuff.

Vogt's theorem on categories of homotopy coherent diagrams

1986 ◽  
Vol 100 (1) ◽  
pp. 65-90 ◽  
Author(s):  
Jean-Marc Cordier ◽  
Timothy Porter
Keyword(s):  
Small Category ◽  
Obstruction Theory ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Enriched Category ◽  
Simplicial Sets ◽  
Continuous Maps ◽  
Compactly Generated

Let Top be the category of compactly generated topological spaces and continuous maps. The category, Top, can be given the structure of a simplicially enriched category (or S-category, S being the category of simplicial sets). For A a small category, Vogt (in [22]) constructed a category, Coh (A, Top), of homotopy coherent A-indexed diagrams in Top and homotopy classes of homotopy coherent maps, and proved a theorem identifying this as being equivalent to Ho (TopA), the category obtained from the category of commutative A-indexed diagrams by localizing with respect to the level homotopy equivalences. Thus one of the important consequences of Vogt's result is that it provides concrete coherent models for the formal composites of maps and formal inverses of level homotopy equivalences which are the maps in Ho (TopA). The usefulness of such models and in general of Vogt's results is shown in the series of notes [14–17] by the second author in which those results are applied to give an obstruction theory applicable in prohomotopy theory.

Homotopy of Natural Transformations

1970 ◽  
Vol 22 (2) ◽  
pp. 332-341 ◽  
Author(s):  
K. A. Hardie
Keyword(s):  
Full Subcategory ◽  
Natural Transformation ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Base Points ◽  
Natural Transformations ◽  
Image Position

Let C be a full subcategory of T, the category of based topological spaces and based maps, and let Cn be the corresponding category of n-tuples. Let S, T: Tn → T be covariant functors which respect homotopy classes and let u, v: S → T be natural transformations, u and v are homotopic inC, denoted u ≃ v(C), if uX ≃ vX: SX → TX (X ∈ Cn), that is to say, for every X ∈ C, uX and vX are homotopic (all homotopies are required to respect base points), u and v are naturally homotopic inC, denoted u ≃n v; (C), if there exist morphismssuch that, for every X ∈ C, utX is a homotopy from uX to vX and such that, for every t ∈ I, ut:S → T is a natural transformation.

Universal Bundles and Classifying Spaces

2020 ◽  
pp. 39-44
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Group ◽  
Equivariant Cohomology ◽  
Total Space ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Continuous Maps ◽  
Cw Complexes

This chapter evaluates universal bundles and classifying spaces. As before, G is a topological group. In defining the equivariant cohomology of a G-space M, one needs a weakly contractible space EG on which G acts freely. Such a space is provided by the total space of a universal G-bundle, a bundle from which every principal G-bundle can be pulled back. The base BG of a universal G-bundle is called a classifying space for G. By Whitehead's theorem, for CW-complexes, weakly contractible is the same as contractible. In the category of CW complexes (with continuous maps as morphisms), a principal G-bundle whose total space is contractible turns out to be precisely a universal G-bundle.

scholarly journals Classifying spaces for commutativity of low-dimensional Lie groups

2019 ◽  
Vol 169 (3) ◽  
pp. 433-478 ◽  
Author(s):  
OMAR ANTOLÍN–CAMARENA ◽  
SIMON PHILIPP GRITSCHACHER ◽  
BERNARDO VILLARREAL
Keyword(s):  
Lie Groups ◽  
Homotopy Type ◽  
Steenrod Algebra ◽  
Homotopy Groups ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Cohomology Rings ◽  
Homotopy Fiber ◽  
Low Dimensional

AbstractFor each of the groups G = O(2), SU(2), U(2), we compute the integral and $\mathbb{F}_2$-cohomology rings of BcomG (the classifying space for commutativity of G), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of EcomG (the homotopy fiber of the inclusion BcomG → BG), and some low-dimensional homotopy groups of BcomG.

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

scholarly journals Entropy on noncompact sets

2019 ◽  
Vol 7 (1) ◽  
pp. 29-37
Author(s):  
Jose S. Cánovas
Keyword(s):  
Topological Entropy ◽  
Topological Spaces ◽  
Continuous Maps

AbstractIn this paper we review and explore the notion of topological entropy for continuous maps defined on non compact topological spaces which need not be metrizable. We survey the different notions, analyze their relationship and study their properties. Some questions remain open along the paper.

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