Generalization of Braids and Homologies

Abstract

In this paper, we give a survey of recent results devoted to the homology of generalizations of braids: the homological properties of virtual braids and the generalized homology of Artin groups studied by C. Broto and the author. Virtual braid groups VBn correspond to virtual knots in the same way that classical braids correspond to usual knots. Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. The Burau representation to GLnℤ[t, t−1] is extended from classical braids to virtual ones. Its homological properties are also studied. The following splitting of infinite loop spaces for the plus-construction of the classifying space of the virtual braid group on an infinite number of strings exists: $$\mathbb{Z} \times BV{\kern 1pt} B_\infty ^ + \simeq \Omega ^\infty S^\infty \times S^1 \times Y,$$ where Y is an infinite loop space. Connections with K*ℤ are discussed. In the last section, information on Morava K-theory and the Brown-Peterson homology of Artin groups and braid groups in handlebodies is collected.