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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.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)

Citations:

Zbl 0611.57008
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References:

[1] R Fenn, C Rourke, On Kirby’s calculus of links, Topology 18 (1979) 1 · Zbl 0413.57006 · doi:10.1016/0040-9383(79)90010-7
[2] M H Freedman, F Quinn, Topology of 4-manifolds, Princeton Mathematical Series 39, Princeton University Press (1990) · Zbl 0705.57001
[3] S Garoufalidis, M Goussarov, M Polyak, Calculus of clovers and finite type invariants of 3-manifolds, Geom. Topol. 5 (2001) 75 · Zbl 1066.57015 · doi:10.2140/gt.2001.5.75
[4] S Garoufalidis, J Levine, On finite type 3-manifold invariants II, Math. Ann. 306 (1996) 691 · Zbl 0889.57016 · doi:10.1007/BF01445272
[5] S Garoufalidis, T Ohtsuki, On finite type 3-manifold invariants III: Manifold weight systems, Topology 37 (1998) 227 · Zbl 0889.57017 · doi:10.1016/S0040-9383(97)00028-1
[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 · doi:10.1016/S0764-4442(00)80053-1
[7] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1 · Zbl 0941.57015 · doi:10.2140/gt.2000.4.1
[8] R C Kirby, L R Taylor, A survey of 4-manifolds through the eyes of surgery, Ann. of Math. Stud. 149, Princeton Univ. Press (2001) 387 · Zbl 0974.57012
[9] J P Levine, Surgery on links and the \(\overline\mu\)-invariants, Topology 26 (1987) 45 · Zbl 0611.57008 · doi:10.1016/0040-9383(87)90020-6
[10] T T Q Le, J Murakami, T Ohtsuki, On a universal perturbative invariant of 3-manifolds, Topology 37 (1998) 539 · Zbl 0897.57017 · doi:10.1016/S0040-9383(97)00035-9
[11] S V Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987) 268, 345 · Zbl 0634.57006
[12] H Murakami, Y Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989) 75 · Zbl 0646.57005 · doi:10.1007/BF01443506
[13] T Ohtsuki, Finite type invariants of integral homology 3-spheres, J. Knot Theory Ramifications 5 (1996) 101 · Zbl 0942.57009 · doi:10.1142/S0218216596000084
[14] A Ranicki, The algebraic theory of surgery I: Foundations, Proc. London Math. Soc. \((3)\) 40 (1980) 87 · Zbl 0471.57010 · doi:10.1112/plms/s3-40.1.87
[15] C T C Wall, Surgery on compact manifolds, London Mathematical Society Monographs 1, Academic Press (1970) · Zbl 0219.57024
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