Geometry & Topology

Gauge theoretic invariants of Dehn surgeries on knots

Hans U Boden, Christopher M Herald, Paul A Kirk, and Eric P Klassen

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New methods for computing a variety of gauge theoretic invariants for homology 3–spheres are developed. These invariants include the Chern–Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. These quantities are calculated for flat SU(2) connections on homology 3–spheres obtained by 1k Dehn surgery on (2,q) torus knots. The methods are then applied to compute the SU(3) gauge theoretic Casson invariant (introduced in [J. Diff. Geom. 50 (1998) 147-206]) for Dehn surgeries on (2,q) torus knots for q=3,5,7 and 9.

Article information

Geom. Topol., Volume 5, Number 1 (2001), 143-226.

Received: 20 September 1999
Accepted: 7 March 2001
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 53D12: Lagrangian submanifolds; Maslov index 58J28: Eta-invariants, Chern-Simons invariants 58J30: Spectral flows

homology 3–sphere gauge theory 3–manifold invariants spectral flow Maslov index


Boden, Hans U; Herald, Christopher M; Kirk, Paul A; Klassen, Eric P. Gauge theoretic invariants of Dehn surgeries on knots. Geom. Topol. 5 (2001), no. 1, 143--226. doi:10.2140/gt.2001.5.143.

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  • S Akbulut, J McCarthy, Casson's invariant for oriented homology 3–spheres, an exposition, Mathematical Notes no. 36, Princeton University Press (1990)
  • D Auckly, A topological method to compute spectral flow, Kyungpook Math. J. 38 (1998) 181–203
  • M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry. I,II,III, Math. Proc. Camb. Phil. Soc. 77 (1975) 43–69; 78 (1975) 405–432; 79 (1976) 71–99
  • H U Boden, Unitary representations of Brieskorn spheres, Duke J. Math. 75 (1994) 193–220
  • H U Boden, C M Herald, The SU(3) Casson invariant for integral homology 3–spheres, J. Diff. Geom. 50 (1998) 147–206
  • B Booss-Bavnbek, K Wojciechowski, Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston (1993)
  • U Bunke, On the gluing problem for the $\eta$-invariant, J. Diff. Geom. 41 (1995) 397-448
  • S Cappell, R Lee, E Miller, A symplectic geometry approach to generalized Casson's invariant, Bull. Amer. Math. Soc. 22 (1990) 269–275
  • S Cappell, R Lee, E Miller, On the Maslov index, Comm. Pure Appl. Math. 47 (1994) 121–186
  • S Cappell, R Lee, E Miller Self-adjoint elliptic operators and manifold decompositions: I. Low eigenmodes and stretching, Comm. Pure Appl. Math. 49 (1996) 825–866; II. Spectral flow and Maslov index, 49 (1996) 869–909
  • H S M Coxeter, W O Moser, Generators and Relations for Discrete Groups, 2nd edition, Ergeb. u. Ihrer Grenz. Springer–Verlag, New York (1965)
  • M Daniel, An extension of a theorem of Nicolaescu on spectral flow and the Maslov index, Proc. Amer. Math. Soc. 128 (2000) 611–619
  • M Daniel, Maslov index, symplectic reduction in a symplectic Hilbert space and a splitting formula for spectral flow, PhD Thesis, Indiana University, Bloomington, 1997
  • M Daniel, P Kirk, A general splitting theorem for spectral flow, with an appendix by K P Wojciechowski, Michigan Math. J. 46 (1999) 589–617
  • M Farber, J Levine, Jumps of the eta-invariant. With an appendix by S Weinberger: Rationality of $\varrho$–invariants. Math. Zeit. 223 (1996) 197–246
  • B Fine, P Kirk, E Klassen, A local analytic splitting of the holonomy map on flat connections, Math. Ann. 299 (1994) 171–189
  • R Fintushel, R Stern, Instanton homology of Seifert-fibered 3–spheres, Proc. Lond. Math. Soc. (3) 61 (1990) 109–138
  • T Kato, Perturbation Theory of Linear Operators, 2nd edition, Grund. der math. Wissen. 132, Springer, Berlin (1980)
  • P Kirk, E Klassen, Chern–Simons invariants of 3–manifolds and representation spaces of knot groups, Math. Ann. 287 (1990) 347–367
  • P Kirk, E Klassen, Computing spectral flow via cup products, J. Diff. Geom. 40 (1994) 505–562
  • P Kirk, E Klassen, Analytic deformations of the spectrum of a family of Dirac operators on an odd-dimensional manifold with boundary, Mem. Amer. Math. Soc. 124 (1996) no. 592
  • P Kirk, E Klassen, The spectral flow of the odd signature operator and higher Massey products, Math. Proc. Camb. Phil. Soc. 121 (1997) 297–320
  • P Kirk, E Klassen, Continuity and analyticity of families of self-adjoint Dirac operators on a manifold with boundary, Illinois J. Math. 42 (1998) 123–138
  • P Kirk, E Klassen, D Ruberman, Splitting the spectral flow and the Alexander matrix, Comm. Math. Helv. 69 (1994) 375–416
  • P Kirk, M Lesch, The eta-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary, preprint (2000) arxiv:math.DG/0012123
  • E Klassen, Representations of knot groups in SU(2), Trans. Amer. Math. Soc. 326 (1991) 795–828
  • X-S Lin, Z Wang, Fermat limit and congruence of Ohtsuki invariants, from: “Proceedings of the Kirbyfest (Berkeley, CA, 1998)”, Geometry and Topology Monographs 2 (1999) 321–333
  • T Mrowka, K Walker, private communication of unpublished research (1993)
  • L Nicolaescu, The Maslov index, the spectral flow, and splittings of manifolds, Duke Math. J. 80 (1995) 485–533
  • L Nicolaescu, Generalized symplectic geometries and the index of families of elliptic problems, Mem. Amer. Math. Soc. 126 (1997) no. 609
  • C Taubes, Casson's invariant and gauge theory, J. Diff. Geom. 31 (1990) 547–599
  • K Walker, An extension of Casson's invariant, Annals of Math Studies 126, Princeton University Press (1992)