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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Spin Borromean surgeries

Author: Gwénaël Massuyeau
Journal: Trans. Amer. Math. Soc. 355 (2003), 3991-4017
MSC (2000): Primary 57M27; Secondary 57R15
Published electronically: June 24, 2003
MathSciNet review: 1990572
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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

Keywords: 3-manifolds, finite type invariants, spin structures, $Y$-graphs
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
Article copyright: © Copyright 2003 American Mathematical Society