Floer Homology for Knots and $\text{SU(2)}$ -Representations for Knot Complements and Cyclic Branched Covers

Collin, Olivier

Canadian journal of mathematics, Tome 52 (2000) no. 2, pp. 293-305

Résumé

In this article, using 3-orbifolds singular along a knot with underlying space a homology sphere ${{Y}^{3}}$ , the question of existence of non-trivial and non-abelian $\text{SU(2)}$ -representations of the fundamental group of cyclic branched covers of ${{Y}^{3}}$ along a knot is studied. We first use Floer Homology for knots to derive an existence result of non-abelian $\text{SU(2)}$ -representations of the fundamental group of knot complements, for knots with a non-vanishing equivariant signature. This provides information on the existence of non-trivial and non-abelian $\text{SU(2)}$ -representations of the fundamental group of cyclic branched covers. We illustrate the method with some examples of knots in ${{S}^{3}}$ .

Détail

DOI : 10.4153/CJM-2000-013-6

Mots-clés : 57M57, 57M12, 57M25, 57M05

Bibliographie

[AM] [AM] Akbulut, S. and McCarthy, J., Casson's Invariant for Oriented Homology 3-spheres. Princeton University Press, New Jersey, 1990. Google Scholar

[BM] [BM] Bass, H. and Morgan, J., The Smith Conjecture. Academic Press, New York, 1984. Google Scholar

[BZ] [BZ] Boileau, M. and Zimmermann, B., The π-orbifold Group of a Link. Math. Z. 200(1989), 187–208. Google Scholar

[Bur] [Bur] Burde, G., SU(2)-representation spaces for 2-bridge knot groups. Math. Ann. 288(1990), 103–119. Google Scholar

[Co1] [Co1] Collin, O., Gauge Theory for 3-orbifolds and Knots. D. Phil. Thesis, University of Oxford, 1997. Google Scholar

[Co2] [Co2] Collin, O., Floer Homology for Orbifolds and Gauge Theory Knot Invariants. In: Proceedings of Knots ‘96, World Scientific Publishing Co., Singapore, 1997, 201–212. Google Scholar

[CS] [CS] Collin, O. and Steer, B., Instanton Floer Homology for Knots via 3-orbifolds. J. Differential. Geom., to appear. Google Scholar

[CL] [CL] Cooper, D. and Long, D., Representing Knot Groups into SL(2, C). Proc. Amer.Math. Soc. 116(1992), 547–549. Google Scholar

[EN] [EN] Eisenbud, D. and Neumann, W., Three-dimensional Link Theory and Invariants of Plane Curve Singularities. Princeton University Press, Princeton, 1985. Google Scholar

[FS] [FS] Fintushel, R. and Stern, R., Instanton homology of Seifert fibred homology 3-spheres. Proc. London Math. Soc. 61(1990), 109–137. Google Scholar

[Flo] [Flo] Floer, A., An Instanton Invariant for 3-manifolds. Comm. Math. Phys. 118(1988), 215–240. Google Scholar

[Fox] [Fox] Fox, R., The Homology Characters of the Cyclic Coverings of the Knots of Genus One.. Ann. of Math. 71(1960), 187–196. Google Scholar

[He1] [He1] Herald, C., Legendrian Cobordism and Chern-Simons Theory for 3-manifolds with Boundary. Comm. Anal. Geom. 2(1994), 337–413. Google Scholar

[He2] [He2] Herald, C., Existence of Irreducible Representations ofHomology Knot ComplementsWith Non-constant Equivariant Signature. Math. Ann. 309(1997), 21–35. Google Scholar

[HK] [HK] Heusener, M. and Kroll, M., Deforming Abelian SU(2)-representations of Knot Groups. Comment. Math. Helv. 73(1998), 480–498. Google Scholar

[Kir] [Kir] Kirby, R., Problems in Low-Dimensional Topology. In: Geometric Topology—Proceedings of the 1993 Georgia International Topology Conference 2, International Press, Cambridge, MA, 1997, 35–473. Google Scholar

[KK] [KK] Kirk, P. and Klassen, E., Chern-Simons Invariants for 3-manifolds and Representation Spaces of Knot Groups. Math. Ann. 287(1990), 343–367. Google Scholar

[Kla] [Kla] Klassen, E., Representation Spaces of Knot Groups in SU(2). Trans. Amer. Math. Soc. 326(1991), 795–828. Google Scholar

[Le] [Le] Le, D., Sur les Noeuds Algébriques. Comp. Math. 25(1972), 281–321. Google Scholar

[Li1] [Li1] Litherland, R., Signatures of Iterated Torus Knots. In: Topology of Low-dimensional Manifolds, Proceedings, Sussex 1977, Lecture Notes in Math. 722, Springer-Verlag, Berlin, 1979, 71–84. Google Scholar

[Li2] [Li2] Litherland, R., Cobordism of Satellite Knots. Contemp.Math. 35(1984), 327–362. Google Scholar

[Rat] [Rat] Ratcliffe, J., Foundations of Hyperbolic Manifolds. Springer-Verlag, New York, 1994. Google Scholar

[Rol] [Rol] Rolfsen, D., Knots and Links. Publish or Perish, Houston, 1990. Google Scholar