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49
ARTIN BRAID GROUPS AND HOMOTOPY GROUPS
"... Abstract. We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as a ..."
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Abstract. We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as a summand of the center of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. These results give some deep and fundamental connections between the braid groups and the general higher homotopy groups of spheres. 1.
DÉCALAGE AND KAN’S SIMPLICIAL LOOP GROUP FUNCTOR
"... Abstract. Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these simplicial sets, and it is a theorem due to Cegarra and Remedios an ..."
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Cited by 5 (3 self)
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Abstract. Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these simplicial sets, and it is a theorem due to Cegarra and Remedios and independently Joyal and Tierney, that this comparison map is a weak homotopy equivalence for any bisimplicial set. In this paper we will give a new, elementary proof of this result. As an application, we will revisit Kan’s simplicial loop group functor G. We will give a simple formula for this functor, which is based on a factorization, due to Duskin, of Eilenberg and Mac Lane’s classifying complex functor W. We will give a new, short, proof of Kan’s result that the unit map for the adjunction G ⊣ W is a weak homotopy equivalence for reduced simplicial sets. 1.
Configurations, Braids and HOMOTOPY GROUPS
"... Simplicial and ∆structures of configuration spaces are investigated. New connections between the homotopy groups of the 2sphere and the braid groups are given. The higher homotopy groups of the 2 sphere are shown to be derived groups of the braid groups over the 2sphere. Moreover the higher hom ..."
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Cited by 5 (4 self)
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Simplicial and ∆structures of configuration spaces are investigated. New connections between the homotopy groups of the 2sphere and the braid groups are given. The higher homotopy groups of the 2 sphere are shown to be derived groups of the braid groups over the 2sphere. Moreover the higher homotopy groups of the 2sphere are shown to be isomorphic to the
A Simplicial Description Of The Homotopy Category Of Simplicial Groupoids
, 2000
"... . In this paper we use Quillen's model structure given by DwyerKan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then chara ..."
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. In this paper we use Quillen's model structure given by DwyerKan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characterize the associated left and right homotopy relations in terms of simplicial identities and give a simplicial description of the homotopy category of simplicial groupoids. Finally, we show loop and suspension functors in the pointed case. 1. Introduction 1.1. Summary. A wellknown and quite powerful context in which an abstract homotopy theory can be developed is supplied by a category with a closed model structure in the sense of Quillen [16]. The category Simp(Gp) of simplicial groups is a remarkable example of what a closed model category is, and the homotopy theory in Simp(Gp) developed by Kan [12] occurs as the homotopy theory associated to this closed model structure. According to the t...
Resolutions of spaces by cubes of fibrations
 Hagen Germany T. Porter School of Mathematics University of Wales Bangor Bangor
, 1986
"... J.L. Loday has used wcubes of fibrations, where n is a nonnegative integer, in his study of spaces with finitely many nontrivial homotopy groups [4]. His main result is the construction of an algebraic category equivalent to the weak homotopy category of pathconnected spaces Z with TI^Z = 0 for ..."
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J.L. Loday has used wcubes of fibrations, where n is a nonnegative integer, in his study of spaces with finitely many nontrivial homotopy groups [4]. His main result is the construction of an algebraic category equivalent to the weak homotopy category of pathconnected spaces Z with TI^Z = 0 for /> w+1 [4, 1.7]. One step in
COMBINATORIAL DESCRIPTION OF THE HOMOTOPY GROUPS OF WEDGE OF SPHERES
"... Abstract. In this paper, we give a combinatorial description of the homotopy groups of a wedge of spheres. This result generalizes that of J. Wu on the homotopy groups of a wedge of 2spheres. In particular, the higher homotopy groups of spheres are given as the centers of certain combinatorially de ..."
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Abstract. In this paper, we give a combinatorial description of the homotopy groups of a wedge of spheres. This result generalizes that of J. Wu on the homotopy groups of a wedge of 2spheres. In particular, the higher homotopy groups of spheres are given as the centers of certain combinatorially described groups with special generators and relations. 1.
A case study of A∞structure
 Georgian Mathematical Journal
, 2010
"... The notion of A∞object is now classical. Initiated by Jim Stasheff [29], it can be applied to most algebraic structures, when the required algebraic properties are only satisfied up to homotopy, which homotopy in turn must satisfy some properties, only up to homotopy and so on. The A∞structures ha ..."
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The notion of A∞object is now classical. Initiated by Jim Stasheff [29], it can be applied to most algebraic structures, when the required algebraic properties are only satisfied up to homotopy, which homotopy in turn must satisfy some properties, only up to homotopy and so on. The A∞structures have in particular