. In this paper we use Quillen's model structure given by Dwyer-Kan 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 well-known 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...

Introduction \Gamma-spaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on Kan-Thurston theorem we show that any \Gamma-space is stably weak equivalent to a discrete \Gamma-group. By a well-known theorem of Dold-Kan the Moore normalization establishes the equivalence between the category of simplicial abelian groups and the category of chain complexes (see [DP]). mimicking the construction of normalization of simplicial groups, we give a similar construction for \Gamma-groups. This construction is based on the notion of cross-effects of functors [BP], which is a generalizatin of the classical definition of Eilenberg and Mac Lane [EM] to the non-abelian setup. Finally a Dold-Kan type theorem for the category of \Gamma-groups is proved. In abelian case our theorem claims that the category of abelian \Gamma-groups is equivalent to the category of functors Ab\Omega , where\Om

Abstract. In 1979 Cohen, Moore, and Neisendorfer determined the decomposition into indecomposable pieces, up to homotopy, of the loop space on the mod p Moore space for primes p> 2 and used the results to find the best possible exponent for the homotopy groups of the spheres and for the Moore spaces at such primes. The corresponding problems for p = 2 are still open. In this paper we reduce to algebra the determination of the base indecomposable factor in the decomposition of the mod 2 Moore space. The algebraic problems involved in determining detailed information about this factor are formidable, related to deep unsolved problems in the modular representation theory of the symmetric groups. Our decomposition has not led (thus far) to a proof of the conjectured existence of an exponent for the homotopy groups of the mod 2 Moore space or to an improvement in the known bounds for the exponent of the 2-torsion in the homotopy groups of spheres. 1.

A. Joyal [J] has introduced the category D of the so-called finite disks, and used it to define the concept of #-category, a notion of weak #-category. We introduce the notion of an #-graph being composable (meaning roughly that 'it has a unique composite'), and call an #-category simple if it is freely generated by a composable #-graph. The category S of simple #-categories is a full subcategory of the category, with strict #-functors as morphisms, of all #-categories. The category S is a key ingredient in another concept of weak #-category, called protocategory [MM1], [MZ]. We prove that D and S are contravariantly equivalent, by a duality induced by a suitable schizophrenic object living in both categories. In [MZ], this result is one of the tools used to show that the concept of #-category and that of protocategory are equivalent in a suitable sense. We also prove that composable #-graphs coincide with the #-graphs of the form T # considered by M.Batanin [B], which were characterized by R. Street (as announced in [S]) and called `globular cardinals'. Batanin's construction, using globular cardinals, of the free #-category on a globular set plays an important role in our paper. We give a self-contained presentation of Batanin's construction that suits our purposes.

A functor from simplicial algebras to crossed n-cubes is shown to be an embedding on a reflexive subcategory of the category of simplicial algebras that contains representatives for all n types.

Abstract not found

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 2-spheres. In particular, the higher homotopy groups of spheres are given as the centers of certain combinatorially described groups with special generators and relations. 1.

Abstract. Simplicial and ∆-structures of configuration spaces are investigated. New connections between the homotopy groups of the 2-sphere 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 2-sphere. Moreover the higher homotopy groups of the 2-sphere are shown to be isomorphic to the

Foliated norms on fundamental group and homology

Foliated norms on fundamental group and homology