Homological Properties of Generalizations of Braids
by
Vladimir V. Vershinin
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

Homology of classical braid groups were studied by V. I. Arnold, D. B. Fuks, F. Cohen, G. Segal and others. Recent developments of Low-Dimensional Topology gave rise to various generalizations of braids. We consider the following of them: the braid group in a handlebody Br^g_n, the braid-permutation group BP_n, the virtual braid group VB_n and the singular braid group SG_n.

There exists the following homomorphism:

VBn --> BPn.

Proposition 1 The classical braid group Br_n is a subgroup of the virtual braid group VB_n

Theorem 1 The Quillen's plus-construction of the classifying space of the braid group on the infinite number of strings in the handlebody of the genus g is equivalent to the following product of loop spaces over the spheres

BBr\inftyg+ =~ \Omega2 S3 ×\OmegaS2 × ... ×\OmegaS2,

where the number of factors in the product is equal to the genus g.

Theorem 2 The classifying space of the braid-permutation group on the infinite number of strings after the plus construction becomes an infinite loop space. There exists an infinite loop space Y such that there is an equivalence of the following infinite loop spaces:

BBP+ =~ B\Sigma+\infty ×S1 ×Y.

The same is true for the virtual braids:

BVB+ =~ B\Sigma+\infty ×S1 ×W.

Theorem 3 The classifying space of the singular braid group SG_\infty on the infinite number of strings after the plus construction becomes a loop space. There exists a loop space Z such that there is an equivalence of the following loop spaces:

BSG+ =~ S1 ×BBr+\infty ×Z.

Date received: July 2, 1999