Simplicial homotopy theory, link homology and Khovanov homology

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.

Lie dimension subrings

We compare, for [Formula: see text] a Lie ring over the integers, its lower central series [Formula: see text] and its dimension series defined by [Formula: see text] in the universal enveloping algebra of [Formula: see text]. We show that [Formula: see text] for all [Formula: see text], but give an example showing that they may differ if [Formula: see text]. We introduce simplicial methods to describe these results, and to serve as a possible tool for further study of the dimension series.

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.