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Homology surgery and invariants of 3-manifolds. (English) Zbl 1009.57022
For \(N\) a closed oriented \(3\)-manifold with fundamental group \(\pi\), the authors set up a surgery theory for the set of Z\(\pi\)-homology equivalences from other closed oriented \(3\)-manifolds to \(N\) modulo diffeomorphism. Very roughly, the resulting homology surgery obstruction map takes values in a set of congruence classes of certain non-singular Hermitian matrices. Analogous to the way all closed \(3\)-manifolds are identified with framed links in \(S^3\) modulo the Kirby calculus, there is a link description of this homology surgery theory. In this way the kernel of the surgery map leads to the class of \(\pi\)-algebraically split links in \(N\) and to finite type invariants. Ideas and results in [S. Garoufalidis, M. Goussarov and M. Polyak, Calculus of clovers and finite type invariants of 3-manifolds, Geom. Topol. 5, 75-108 (2001)] concerning finite type invariants of homology \(3\)-spheres based on surgery on algebraically split links are extended to closed orientable \(3\)-manifolds and \(\pi\)-algebraically split links. Milnor type invariants are shown to classify surgery equivalence, generalizing [J. Levine, Topology 26, 45-61 (1987; Zbl 0611.57008)]. Finally, concordant links are shown to be surgery equivalent by a direct geometric argument.

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
Full Text: DOI EMIS EuDML arXiv
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