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Families of SU(2) representations for mapping cylinders of periodic monodromy

Published online by Cambridge University Press:  20 January 2009

G. Daskalopoulos ,
S. Dostoglou  and
R. Wentworth
G. Daskalopoulos
Affiliation:
Department of Mathematics, Brown University, Providence, RI 02912, U.S.A.
S. Dostoglou
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.
R. Wentworth
Affiliation:
Department of Mathematics, University of California at Irvine, Irvine, CA 92717, U.S.A.
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Abstract

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We examine the action of diffeomorphisms of an oriented surface with boundary on the space of conjugacy classes of SU(2) representations of the fundamental group and prove that in the case of a single periodic diffeomorphism the induced action always has fixed points. For the corresponding 3-dimensional mapping cylinders we obtain families of representations parametrized by their value on the longitude of the torus boundary.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 2 , June 1997 , pp. 383 - 392
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

REFERENCES

1. Culler, M., Gordon, C. Mca., Luecke, J., and Shalen, P., Dehn surgery on knots, Ann. of Math. 125, (1987), 237–300.CrossRefGoogle Scholar
2. Daskalopoulos, G. and Wentworth, R., Geometric quantization for the moduli space of vector bundles with parabolic structure, unpublished.Google Scholar
3. Dostoglou, S. and Salamon, D., Instanton homology and symplectic fixed points, (London Math. Soc. Lecture Notes 192, 1994), 57–93.Google Scholar
4. Frankel, T., Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165–174.CrossRefGoogle Scholar
5. Frohman, C., Unitary representations of knot groups, Topology 32 (1993), 121–144.Google Scholar
6. Frohman, C. and Long, D., Casson's invariant and surgery on knots, Proc. Edinburgh Math. Soc. 35 (1992), 383–395.CrossRefGoogle Scholar
7. Kerckhoff, S., The Nielsen realization problem, Ann. of Math. 117 (1983), 235–265.Google Scholar
8. Moser, L., Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745.CrossRefGoogle Scholar
9. Mehta, V. and Seshadri, C., Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205–239.CrossRefGoogle Scholar
10. Narasimhan, M. S. and Ramanan, S., Geometry of Hecke cycles -I, in C. P. Ramanujam: A Tribute, (Springer-Verlag, New York, 1978).Google Scholar
11. Seifert, H., Uber das Geschlecht von Knoten, Math. Ann. 110 (1934), 571–592.CrossRefGoogle Scholar
12. Seshadri, C., Fibrés vectoriels sur les courbes algébriques, Astérisque 96 (1982).Google Scholar