# zbMATH — the first resource for mathematics

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)
Full Text:
##### 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 · emis:journals/UW/gt/GTVol5/paper3.abs.html · eudml:124859 · arxiv:math/0005192 [4] S Garoufalidis, J Levine, On finite type 3-manifold invariants II, Math. Ann. 306 (1996) 691 · Zbl 0889.57016 · doi:10.1007/BF01445272 · eudml:165475 [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 · emis:journals/UW/gt/GTVol4/paper1.abs.html · eudml:120433 · arxiv:math/0001185 [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 · eudml:164541 [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
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.