zbMATH — the first resource for mathematics

Cohomology rings, Rochlin function, linking pairing and the Goussarov-Habiro theory of three-manifolds. (English) Zbl 1056.57009
Among compact oriented \(3\)-manifolds, \(Y_k\)-equivalence is the equivalence relation generated by surgeries along graph claspers of degree \(k\) for any integer \(k\geq 1\), introduced by M. Goussarov [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 6, 517–522 (1999; Zbl 0938.57013)] and K. Habiro [Geom. Topol. 4, 1–83 (2000; Zbl 0941.57015)], independently.
This paper deals with the problem of characterizing the \(Y_k\)-equivalence relation in terms of invariants of the manifolds. For \(k=1\), S. V. Matveeev [Math. Notes 42, No. 1/2, 651–656 (1987; Zbl 0649.57010); translation from Mat. Zametki 42, No. 2, 268–278 (1987)] showed that two closed oriented \(3\)-manifolds are \(Y_1\)-equivalent if and only if they have isomorphic pairs (homology, linking pairing). For \(k=2\), the problem was solved for a certain class of manifolds with boundary by G. Massuyeau and J.-B. Meilhan [J. Knot Theory Ramifications 12, No. 4, 493–522 (2003; Zbl 1065.57024)].
The paper under review deals with the \(Y_2\)-equivalence for closed \(3\)-manifolds. It proves that two closed oriented \(3\)-manifolds are \(Y_2\)-equivalent if and only if they have isomorphic quintuplets (homology, space of spin structures, linking pairing, cohomology rings, Rochlin function). The spin case and the complex spin case are also considered.

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57N10 Topology of general \(3\)-manifolds (MSC2010)
Full Text: DOI EMIS EuDML arXiv
[1] T D Cochran, A Gerges, K Orr, Dehn surgery equivalence relations on 3-manifolds, Math. Proc. Cambridge Philos. Soc. 131 (2001) 97 · Zbl 0984.57010
[2] T D Cochran, P Melvin, Finite type invariants of 3-manifolds, Invent. Math. 140 (2000) 45 · Zbl 0949.57010
[3] F Deloup, G Massuyeau, Quadratic functions and complex spin structures on three-manifolds, Topology 44 (2005) 509 · Zbl 1071.57009
[4] S Garoufalidis, M Goussarov, M Polyak, Calculus of clovers and finite type invariants of 3-manifolds, Geom. Topol. 5 (2001) 75 · Zbl 1066.57015
[5] C Gille, Sur certains invariants récents en topologie de dimension 3, Thèse de Doctorat, Université de Nantes (1998)
[6] M Goussarov, Finite type invariants and \(n\)-equivalence of 3-manifolds, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 517 · Zbl 0938.57013
[7] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1 · Zbl 0941.57015
[8] D Johnson, The structure of the Torelli group III: The abelianization of \(\mathcalT\), Topology 24 (1985) 127 · Zbl 0571.57010
[9] D Johnson, A survey of the Torelli group, Contemp. Math. 20, Amer. Math. Soc. (1983) 165 · Zbl 0553.57002
[10] J Lannes, F Latour, Signature modulo 8 des variétés de dimension \(4k\) dont le bord est stablement parallélisé, C. R. Acad. Sci. Paris Sér. A 279 (1974) 705 · Zbl 0303.57011
[11] E Looijenga, J Wahl, Quadratic functions and smoothing surface singularities, Topology 25 (1986) 261 · Zbl 0615.32014
[12] G Massuyeau, Spin Borromean surgeries, Trans. Amer. Math. Soc. 355 (2003) 3991 · Zbl 1028.57017
[13] G Massuyeau, J B Meilhan, Characterization of \(Y_2\)-equivalence for homology cylinders, J. Knot Theory Ramifications 12 (2003) 493 · Zbl 1065.57024
[14] S V Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987) 268, 345 · Zbl 0634.57006
[15] J Milnor, Spin structures on manifolds, Enseignement Math. \((2)\) 9 (1963) 198 · Zbl 0116.40403
[16] J W Morgan, D P Sullivan, The transversality characteristic class and linking cycles in surgery theory, Ann. of Math. \((2)\) 99 (1974) 463 · Zbl 0295.57008
[17] T Ohtsuki, Quantum invariants, Series on Knots and Everything 29, World Scientific Publishing Co. (2002) · Zbl 1163.57302
[18] V G Turaev, Cohomology rings, linking coefficient forms and invariants of spin structures in three-dimensional manifolds, Mat. Sb. \((\)N.S.\()\) 120(162) (1983) 68, 143 · Zbl 0519.57009
[19] C T C Wall, Quadratic forms on finite groups II, Bull. London Math. Soc. 4 (1972) 156 · Zbl 0251.20024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.