Decomposable Free Loop Spaces

1987 ◽  
Vol 39 (4) ◽  
pp. 938-955 ◽  
Author(s):  
J. Aguadé
Keyword(s):  
Topological Group ◽  
Loop Space ◽  
Base Point ◽  
Point Of View ◽  
Homotopy Equivalence ◽  
Compact Open Topology ◽  
Loop Spaces ◽  
Continuous Maps

In this paper we study the spaces X having the property that the space of free loops on X is equivalent in some sense to the product of X by the space of based loops on X. We denote by ΛX the space of all continuous maps from S1 to X, with the compact-open topology. ΩX denotes, as usual, the loop space of X, i.e., the subspace of ΛX formed by the maps from S1 to X which map 1 to the base point of X.If G is a topological group then every loop on G can be translated to the base point of G and the space of free loops ΛG is homeomorphic to G × ΩG. More generally, any H-space has this property up to homotopy. Our purpose is to study from a homotopy point of view the spaces X for which there is a homotopy equivalence between ΛX and X × ΩX which is compatible with the inclusion ΩX ⊂ ΛX and the evaluation map ΛX → X.

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 RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

2021 ◽  
pp. 1-29
Author(s):  
DREW HEARD
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Algebraic Model ◽  
Compact Lie Group ◽  
Loop Spaces ◽  
Local Systems ◽  
Special Case ◽  
Finite Loop ◽  
Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

On the quotients of mapping class groups of surfaces by the Johnson subgroups

2019 ◽  
pp. 1-23
Author(s):  
TOMÁŠ ZEMAN
Keyword(s):  
Boundary Component ◽  
Loop Space ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Classifying Spaces ◽  
Class Groups ◽  
Loop Spaces ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Plus Construction

Abstract We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

scholarly journals M-BRANE MODELS AND LOOP SPACES

2012 ◽  
Vol 27 (20) ◽  
pp. 1230019 ◽  
Author(s):  
CHRISTIAN SÄMANN
Keyword(s):  
Loop Space ◽  
The Self ◽  
String Equation ◽  
Loop Spaces ◽  
Brane Model ◽  
Recent Developments ◽  
Brane Models ◽  
Space Approach

I review an extension of the ADHMN construction of monopoles to M-brane models. This extended construction gives a map from solutions to the Basu–Harvey equation to solutions to the self-dual string equation transgressed to loop space. Loop spaces appear in fact quite naturally in M-brane models. This is demonstrated by translating a recently proposed M5-brane model to loop space. Finally, I comment on some recent developments related to the loop space approach to M-brane models.

On the Homotopy-Commutativity of Loop-Spaces and Suspensions

1968 ◽  
Vol 20 ◽  
pp. 1531-1536 ◽  
Author(s):  
C. S. Hoo
Keyword(s):  
Loop Space ◽  
Loop Spaces

Let X be a space. We are interested in the homotopy-commutativity of the loop-space ΩX and the suspension ΣX, that is, in the question whether or not nil X ≦ 1, conil X ≦ 1, respectively. Let c: ΩX× ΩX ⟶ ΩX, c': ΣX ⟶ ΣX V ΣX be the commutator and co-commutator maps, respectively.

scholarly journals On the geometry of free loop spaces

2002 ◽  
Vol 30 (1) ◽  
pp. 15-23 ◽  
Author(s):  
P. Manoharan
Keyword(s):  
Fundamental Form ◽  
Open Subset ◽  
Loop Space ◽  
Free Loop Space ◽  
Loop Spaces ◽  
Central Extensions ◽  
Christoffel Symbols

We verify the following three basic results on the free loop spaceLM. (1) We show that the set of all points, where the fundamental form onLMis nondegenerate, is an open subset. (2) The connections of a Fréchet bundle overLMcan be extended toS1-central extensions and, in particular, there exist natural connections on the string structures. (3) The notion of Christoffel symbols and the curvature are introduced onLMand they are described in terms of Christoffel symbols ofM.

RACK SPACES AND LOOP SPACES

1999 ◽  
Vol 08 (01) ◽  
pp. 99-114 ◽  
Author(s):  
Bert Wiest
Keyword(s):  
Homotopy Type ◽  
Loop Space ◽  
Topological Spaces ◽  
Loop Spaces

We prove that the rack and quandle spaces of links in 3-manifolds, considered only as topological spaces (disregarding their cubical structure), are closely related to certain subspaces of the loop spaces on the 3-manifold, which we call the vertical and the straight loop space of the link. Using these models we prove that the homotopy type of the non-augmented rack and quandle spaces of a link L in a 3-manifold M depends essentially only on the homotopy type of the pair (M,M -L).

On first countable quasitopological homotopy groups

2021 ◽  
Vol 71 (3) ◽  
pp. 773-779
Author(s):  
Hamid Torabi
Keyword(s):  
Homotopy Group ◽  
Loop Space ◽  
Homotopy Groups ◽  
Compact Open Topology ◽  
Quotient Maps

Abstract If q: X → Y is a quotient map, then, in general, q × q: X × X → Y × Y may fail to be a quotient map. In this paper, by reviewing the concept of homotopy groups and quotient maps, we find under which conditions the map q × q is a quotient map, where q: Ω n (X, x 0) → πn (X, x 0), is the natural quotient map from the nth loop space of (X, x 0), Ω n (X, x 0), with compact-open topology to the quasitopological nth homotopy group πn (X, x 0). Ultimately, using these results, we found some properties of first countable homotopy groups.

Vladimir Abramovich Rokhlin and algebraic topology

2021 ◽  
pp. 33-67
Author(s):  
Victor Buchstaber
Keyword(s):  
Algebraic Topology ◽  
Loop Space ◽  
Vector Bundles ◽  
Point Of View ◽  
Characteristic Classes ◽  
Scientific Heritage ◽  
Content Type ◽  
Modern Development

The article considers the scientific heritage of V. A. Rokhlin in algebraic topology from the point of view of the modern development of mathematics and shows the influence of his results on the development of algebraic topology up to the present. The second part of the article contains new results with fairly detailed sketches of their proofs. There we introduce the notion of partially framed manifolds, which naturally arise in the study of the characteristic classes of vector bundles over the loop space Ω S U ( 2 ) = Ω S P ( 1 ) \Omega SU(2)=\Omega SP(1) . We obtain theorems on the divisibility of the signature of such manifolds as a result of calculations of characteristic classes with values in complex and quaternionic cobordism.

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