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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.

MSC:
 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)
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