On virtual cabling and a structure of 4-strand virtual pure braid group

This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group [Formula: see text]. We define simplicial group [Formula: see text] and its simplicial subgroup [Formula: see text] which is generated by [Formula: see text]. Consequently, we describe [Formula: see text] as HNN-extension and find presentation of [Formula: see text] and [Formula: see text]. As an application to classical braids, we find a new presentation of the Artin pure braid group [Formula: see text] in terms of the cabled generators.

Leibniz Rule on Higher Pages of Unstable Spectral Sequences

A natural composition ⊙ on all pages of the lower central series spectral sequence for spheres is defined. Moreover, it is defined for the p-lower central series spectral sequence of a simplicial group. It is proved that the rth differential satisfies a ‘Leibniz rule with suspension’: dr(a ⊙ σ b) = ±dra ⊙ b + a ⊙ dr σ b, where σ is the suspension homomorphism.

EQUIVALENCE OF MODELS FOR EQUIVARIANT (∞, 1)-CATEGORIES

In this paper, we show that the known models for (∞, 1)-categories can all be extended to equivariant versions for any discrete groupG. We show that in two of the models we can also consider actions of any simplicial groupG.

AN EXPLICIT DESCRIPTION OF THE SIMPLICIAL GROUP

We give an explicit construction for a$K(A, n)$simplicial group and explain its topological interpretation.

CROSSED MODULE BUNDLE GERBES; CLASSIFICATION, STRING GROUP AND DIFFERENTIAL GEOMETRY

We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to a (Lie) crossed module (H → D) there is a simplicial group [Formula: see text], the nerve of the groupoid [Formula: see text] defined by the crossed module, and its geometric realization, the topological group [Formula: see text]. We introduce crossed module bundle gerbes so that their (stable) equivalence classes are in a bijection with equivalence classes of principal [Formula: see text]-bundles. We discuss the string group and string structures from this point of view. Also, we give a simplicial interpretation to the bundle gerbe connection and bundle gerbe B-field.

Simplicial Crossed Modules and Mapping Cones

Given a bisimplicial group 𝐺∗∗ such that 𝑁(𝐺)∗𝑞 = {1} for 𝑞 ⩾ 2, a simplicial group is obtained whose Moore complex is a mapping cone of the chain morphism 𝑁(𝐺)∗1 → 𝑁(𝐺)∗0. This simplicial group is homotopy equivalent to the diagonal of 𝐺∗∗. In the last section a special case is considered.

A BRAIDED SIMPLICIAL GROUP

By studying the braid group action 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 groups. This establishes a relation between the braid groups and the homotopy groups of the sphere.2000Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10.