Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-19T17:24:40.192Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Abe, E.. Hopf Algebras (Cambridge University Press, 1977).Google Scholar
[2]Drinfeld, V. G.. Quantum groups. In Proc. Int. Congress Math., Berkeley 1986, pp. 798–820. (American Mathematical Society, 1987.)Google Scholar
[3]Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millett, K. and Ocneanu, A.. A new polynomial invariant of links. Bull. Amer. Math. Soc. (2) 12 (1985), 239–246.CrossRefGoogle Scholar
[4]Goldman, W.. Invariant functions of Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85 (1986), 263–302.CrossRefGoogle Scholar
[5]Hoste, J. and Kidwell, M.. Dichromatic link invariants. Trans. Amer. Math. Soc. (To appear.)Google Scholar
[6]McCullough, D.. Mappings of reducible 3-manifolds. In Geometric and Algebraic Topology, pp. 61–76, Banach Centre Publications, vol. 18. (PWN-Polish Scientific Publishers, 1986.)Google Scholar
[7]Ozawa, T.. Quantization of a Poisson algebra and polynomials associated to links. (MSRI preprint, 1989.)Google Scholar
[8]Przytycki, J. H.. Skein modules of 3-manifolds. Bull. Polish Acad. Sci. Math. (To appear.)Google Scholar
[9]Przytycki, J. H. and Traczyk, P.. Invariants of links of Conway type. Kobe J. Math. 4 (1987), 115–139.Google Scholar
[10]Turaev, V. G.. Algebras of loops on surfaces, algebras of knots, and quantization. (LOMI preprint, 1988.)Google Scholar
[11]Turaev, V. G.. The Conway and Kauffman modules of the solid torus with an appendix on the operator invariants of tangles. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 167 (1988), 79–89.Google Scholar