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Homotopy skein modules of orientable 3-manifolds

Published online by Cambridge University Press:  24 October 2008

Jim Hoste  and
Jósef H. Przytycki
Jim Hoste
Affiliation:
Pitzer College, Claremont, CA 91711, U.S.A.
Jósef H. Przytycki
Affiliation:
Mathematics Department, University of British Columbia, Vancouver, BC, V6T 1W5, Canada and Warsaw University, Warsaw, Poland
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Abstract

We define the homotopy skein module of an arbitrary orientable 3-manifold M. This module is similar to the ordinary skein module defined by the second author but is more appropriate when considering oriented links in M up to link homotopy rather than isotopy. We compute the homotopy skein module of M = F × I for any orientable surface F and show that it is free. In the case where M = F × I the homotopy skein module may be given an algebra structure and we show that as an algebra it is isomorphic to the universal enveloping algebra of the Goldman–Wolpert Lie algebra of F. We show, also in this case, that the homotopy skein module is a quantization of the symmetric tensor algebra associated to the Goldman–Wolpert Lie algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 3 , November 1990 , pp. 475 - 488
Copyright
Copyright © Cambridge Philosophical Society 1990

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