Geometry & Topology Monographs, Vol. 4 (2002),
Invariants of knots and 3-manifolds (Kyoto 2001),
Paper no. 10, pages 143--160.

Loop spaces of configuration spaces and finite type invariants

Toshitake Kohno

Abstract. The total homology of the loop space of the configuration space of ordered distinct n points in R^m has a structure of a Hopf algebra defined by the 4-term relations if m>2. We describe a relation of between the cohomology of this loop space and the set of finite type invariants for the pure braid group with n strands. Based on this we give expressions of certain link invariants as integrals over cycles of the above loop space.

Keywords. Loop space, configuration space, finite type invariants, braid group, iterated integral

AMS subject classification. Primary: 55P35. Secondary: 20F36, 57M27.

E-print: arXiv:math.GT/0211056

Submitted to GT on 19 December 2001. (Revised 9 April 2002.) Paper accepted 22 July 2002. Paper published 19 September 2002. Corrections to superscripts of B on pages 148 and 149 made on 10 September 2003.

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Toshitake Kohno
Graduate School of Mathematical Sciences
University of Tokyo, Tokyo 153-8914 Japan
Email: kohno@ms.u-tokyo.ac.jp

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