Borromean surgery equivalence of spin 3-manifolds with boundary (Q2792161)

In [Mat. Zametki 42, No. 2, 268--278 (1987; Zbl 0634.57006)], \textit{S. V. Matveev} introduced Borromean surgery and gave the following characterization:NEWLINENEWLINE``Two closed, oriented \(3\)-manifold \(M\) and \(M'\) are related by a sequence of Borromean surgeries if and only if there is an isomorphism \(f: H_1(M, \mathbb{Z}) \to H_1(M', \mathbb{Z})\) inducing isomorphism on the torsion linking pairings.''NEWLINENEWLINE\textit{M. Goussarov} [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 6, 517--522 (1999; Zbl 0938.57013)] and \textit{K. Habiro} [Geom. Topol. 4, 1--83 (2000; Zbl 0941.57015)] considered surgery along so called claspers, which refine the notion of Borromean surgery. One obtains an equivalence relation on \(3\)-manifolds called \(Y\)-equivalence which is generated by \(Y\)-surgeries. Here a \(Y\)-surgery means a surgery along a \(Y\)-clasper.NEWLINENEWLINE\textit{G. Massuyeau} showed in [Trans. Am. Math. Soc. 355, No. 10, 3991--4017 (2003; Zbl 1028.57017)] that Borromean surgery induces a natural correspondence on spin structures.NEWLINENEWLINEThe article under review generalizes these results to compact spin \(3\)-manifolds with boundary:NEWLINENEWLINE``Let \((M, \phi,s)\) and \((M', \phi',s')\) be two \((\Sigma,s_\Sigma)\)-bordered spin \(3\)-manifolds. If NEWLINE\[NEWLINE(M,\phi,s) \text{ and } (M',\phi',s') \text{ are } Y\text{-equivalent}NEWLINE\]NEWLINE then \((M,\phi)\) and \((M',\phi')\) are \(Y\)-equivalent and NEWLINE\[NEWLINER_8((M,\phi,s),(M',\phi',s')) = 0 \text{ (mod 8)}.NEWLINE\]NEWLINE Furthermore, if \(H_1(M, \mathbb{Z})\) has no \(2\)-torsion, then NEWLINE\[NEWLINE(M,\phi) \text{ and } (M', \phi') \text{ are } Y\text{-equivalent},NEWLINE\]NEWLINE implies NEWLINE\[NEWLINE(M,\phi,s) \text{ and } (M',\phi',s') \text{ are } Y\text{-equivalent}.\text{''}NEWLINE\]