Algebraic & Geometric Topology

Splitting the spectral flow and the $\mathrm{SU}(3)$ Casson invariant for spliced sums

Hans U Boden and Benjamin Himpel

Full-text: Open access


We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3–manifolds split along a torus.

Article information

Algebr. Geom. Topol., Volume 9, Number 2 (2009), 865-902.

Received: 8 April 2008
Revised: 1 April 2009
Accepted: 5 April 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J30: Spectral flows
Secondary: 57M27: Invariants of knots and 3-manifolds 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]

gauge theory spectral flow Maslov index spliced sum torus knot


Boden, Hans U; Himpel, Benjamin. Splitting the spectral flow and the $\mathrm{SU}(3)$ Casson invariant for spliced sums. Algebr. Geom. Topol. 9 (2009), no. 2, 865--902. doi:10.2140/agt.2009.9.865.

Export citation


  • M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69
  • H U Boden, C M Herald, The $\rm SU(3)$ Casson invariant for $3$–manifolds split along a $2$–sphere or a $2$–torus, from: “Proceedings of the 1999 Georgia Topology Conference (Athens, GA)”, Topology Appl. 124 (2002) 187–204
  • H U Boden, C M Herald, P A Kirk, An integer valued ${\rm SU}(3)$ Casson invariant, Math. Res. Lett. 8 (2001) 589–603
  • H U Boden, C M Herald, P A Kirk, The integer valued ${\rm SU}(3)$ Casson invariant for Brieskorn spheres, J. Differential Geom. 71 (2005) 23–83
  • H U Boden, C M Herald, P A Kirk, E P Klassen, Gauge theoretic invariants of Dehn surgeries on knots, Geom. Topol. 5 (2001) 143–226
  • S Boyer, A Nicas, Varieties of group representations and Casson's invariant for rational homology $3$–spheres, Trans. Amer. Math. Soc. 322 (1990) 507–522
  • S E Cappell, R Lee, E Y Miller, On the Maslov index, Comm. Pure Appl. Math. 47 (1994) 121–186
  • M Daniel, An extension of a theorem of Nicolaescu on spectral flow and the Maslov index, Proc. Amer. Math. Soc. 128 (2000) 611–619
  • S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Oxford Math. Monog., Oxford Science Publ., The Clarendon Press, Oxford University Press, New York (1990)
  • S Fukuhara, N Maruyama, A sum formula for Casson's $\lambda$–invariant, Tokyo J. Math. 11 (1988) 281–287
  • B Himpel, A splitting formula for the spectral flow of the odd signature operator on $3$–manifolds coupled to a path of ${\rm SU}(2)$ connections, Geom. Topol. 9 (2005) 2261–2302
  • P A Kirk, M Lesch, The $\eta$–invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary, Forum Math. 16 (2004) 553–629
  • E P Klassen, Representations of knot groups in ${\rm SU}(2)$, Trans. Amer. Math. Soc. 326 (1991) 795–828
  • L I Nicolaescu, The Maslov index, the spectral flow, and decompositions of manifolds, Duke Math. J. 80 (1995) 485–533
  • N Saveliev, Invariants for homology $3$–spheres, Low-Dimensional Topology I, Encyclopaedia of Math. Sciences 140, Springer, Berlin (2002)
  • C H Taubes, Casson's invariant and gauge theory, J. Differential Geom. 31 (1990) 547–599