Algebraic & Geometric Topology

Borromean surgery formula for the Casson invariant

Jean-Baptiste Meilhan

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It is known that every oriented integral homology 3–sphere can be obtained from S3 by a finite sequence of Borromean surgeries. We give an explicit formula for the variation of the Casson invariant under such a surgery move. The formula involves simple classical invariants, namely the framing, linking number and Milnor’s triple linking number. A more general statement, for n independent Borromean surgeries, is also provided.

Article information

Algebr. Geom. Topol., Volume 8, Number 2 (2008), 787-801.

Received: 5 February 2008
Revised: 3 March 2008
Accepted: 5 March 2008
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M27: Invariants of knots and 3-manifolds

Casson invariant Borromean surgery finite type invariants


Meilhan, Jean-Baptiste. Borromean surgery formula for the Casson invariant. Algebr. Geom. Topol. 8 (2008), no. 2, 787--801. doi:10.2140/agt.2008.8.787.

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  • E Auclair, C Lescop, Clover calculus for homology $3$–spheres via basic algebraic topology, Algebr. Geom. Topol. 5 (2005) 71–106
  • S Garoufalidis, M Goussarov, M Polyak, Calculus of clovers and finite type invariants of $3$–manifolds, Geom. Topol. 5 (2001) 75–108
  • M Goussarov, Finite type invariants and $n$–equivalence of $3$–manifolds, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 517–522
  • K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1–83
  • J Johannes, A type $2$ polynomial invariant of links derived from the Casson–Walker invariant, J. Knot Theory Ramifications 8 (1999) 491–504
  • C Lescop, Global surgery formula for the Casson–Walker invariant, Annals of Math. Studies 140, Princeton University Press (1996)
  • C Lescop, A sum formula for the Casson–Walker invariant, Invent. Math. 133 (1998) 613–681
  • G Massuyeau, Spin Borromean surgeries, Trans. Amer. Math. Soc. 355 (2003) 3991–4017
  • S V Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987) 268–278, 345
  • Y Mizuma, On the Casson invariant of Mazur's homology spheres, preprint
  • T Ohtsuki, Finite type invariants of integral homology $3$–spheres, J. Knot Theory Ramifications 5 (1996) 101–115
  • K Walker, An extension of Casson's invariant, Annals of Math. Studies 126, Princeton University Press (1992)