Spin Borromean surgeries
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Abstract:
In 1986, Matveev defined the notion of Borromean surgery for closed oriented $3$-manifolds and showed that the equivalence relation generated by this move is characterized by the pair (first Betti number, linking form up to isomorphism).
We explain how this extends for $3$-manifolds with spin structure if we replace the linking form by the quadratic form defined by the spin structure. We then show that the equivalence relation among closed spin $3$-manifolds generated by spin Borromean surgeries is characterized by the triple (first Betti number, linking form up to isomorphism, Rochlin invariant modulo $8$).
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Additional Information
- Gwénaël Massuyeau
- Affiliation: Laboratoire Jean Leray, UMR 6629 CNRS/Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 03, France
- Email: massuyea@math.univ-nantes.fr
- Received by editor(s): April 16, 2001
- Received by editor(s) in revised form: April 2, 2002
- Published electronically: June 24, 2003
- Additional Notes: Commutative diagrams were drawn with Paul Taylor’s package
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 3991-4017
- MSC (2000): Primary 57M27; Secondary 57R15
- DOI: https://doi.org/10.1090/S0002-9947-03-03071-X
- MathSciNet review: 1990572