On loop spaces of configuration spaces, and braid-like groups
by
Fred Cohen
University of Rochester
Coauthors: Sam Gitler

The main topic in this lecture is the structure of loop spaces of classical configuration spaces for k ordered distinct points in a manifold M.

(1) The singular homology of these spaces admit structures which are frequently extensions of the " universal Yang-Baxter Lie algebra " and which depend on the underlying properties of the tangent bundle for M.

(2) There are natural associated groups to these Lie algebras which satisfy properties analogous to those of Artin's braid groups, and are gotten by assembling the images of the classical Hurewicz homomorphism for high dimensional homotopy groups. Furthermore, these groups admit a topological interpretation of "braiding" certain subspaces of a manifold via a "time" parameter.

(3) Some parallels with invariants of pure braids together with their homotopy theoretic interpretations are given. For example, one of the Lie algebras above also arises as the Lie algebra attached to the descending central series for Artin's pure braid group, and thus via Vassiliev invariants of pure braids. There are analogous Lie algebras "with symmetries" constructed by M. Xicotencatl for "orbit configuration spaces".

Date received: June 2, 1999