The maps from loop suspensions to loop spaces are investigated using group representations in this article. The shuffle relations on the Cohen groups are given. By using these relations, a universal ring for functorial self maps of double loop spaces of double suspensions is given. Moreover the obstructions to the classical exponent problem in homotopy theory are displayed in the extension groups of the dual of the important symmetric group modules Lie(n), as well as in the top cohomology of the Artin braid groups with coefficients in the top homology of the Artin pure braid groups.

Abstract. The homotopy theory of the suspensions of the real projective plane is largely investigated. The homotopy groups are computed up to certain range. The decompositions of the self smashes and the loop spaces are studied with some applications to the Stiefel manifolds.

Abstract. The purpose of this article is to describe connections between the loop space of the 2-sphere, and Artin’s braid groups. The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno [19, 20], and provides connections between these various topics. Two consequences are as follows: (1) the homotopy groups of spheres are identified as “natural ” sub-quotients of free products of pure braid groups, and (2) an axiomatization of certain simplicial groups arising from braid groups is shown to characterize the homotopy types of connected CW-complexes. 1. A tale of two groups plus one more In 1924 E. Artin [1, 2] defined the n-th braid group Bn together with the n-th pure braid group Pn, the kernel of the natural map of Bn to Σn the symmetric group on n-letters. It is the purpose of this article to derive additional connections of these groups to homotopy theory, as well as some overlaps with algebraic, and topological properties of braid groups. This article gives certain relationships between free groups on n generators Fn, and braid groups which serve as a bridge between different structures. These connections, at the interface of homotopy groups of spheres, braids, and homotopy links, admit a common thread given by a simplicial group. Recall that a simplicial group Γ ∗ is a collection of groups together with face operations and degeneracy operations

Abstract. By studying braid group actions on Milnor’s construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described group. We also give certain representation of higher differentials in the Adams spectral sequence for π∗(S 2). 1.

this paper is to translate these algebraic results into geometry. Furthermore we will study the homotopy theory of the functorial retracts of #X when X is a p-local two-complex. We start with some observations. We write H (X) for the mod p homology of X

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.

Abstract. In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for braid and surface groups. 1.

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Abstract. In this paper homotopical methods for the description of subgroups determined by ideals in group rings are introduced. It is shown that in certain cases the subgroups determined by symmetric product of ideals in group rings can be described with the help of homotopy groups of spheres. 1.

In this paper, we determine the homotopy groups π4(ΣK(A, 1)) and π5(ΣK(A, 1)) for abelian groups A by using the following methods from group theory and homotopy theory: derived functors, the Carlsson simplicial construction, the Baues-Goerss spectral sequence, homotopy decompositions and the methods of algebraic K-theory. As the applications, we also determine πi(ΣK(G, 1)) with i = 4, 5 for some non-abelian groups G = Σ3 and SL(Z), and π4(ΣK(A4, 1)) for the 4-th alternating group A4.