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On finite type 3-manifold invariants. II. (English) Zbl 0889.57016

The purpose of the present paper is, among other things, to relate the seemingly unrelated notions of surgical equivalence of links in \(S^3\) [J. Levine, Topology 26, 45-61 (1987; Zbl 0611.57008)] and the notion of finite type invariants of oriented integral homology 3-spheres, due to T. Ohtsuki [Finite type invariants of integral homology 3-spheres, J. Knot Theory Ramifications 5, No. 1, 101-115 (1996)]. The paper consists of two parts. In the first part the authors classify pure braids and string links modulo the relation of surgical equivalence. They prove that the group of surgical equivalence classes of pure braids is isomorphic to the corresponding group of string links (Theorem 2). They also give two alternative descriptions of the above mentioned group \(P^{\mathbb{S} \mathbb{E}} (n)\) of surgical equivalence classes of \(n\) strand pure braids: one as a semidirect product of \(P^{\mathbb{S} \mathbb{E}} (n-1)\) together with an explicit quotient of the free group, and another description (Theorem 3) as a group of automorphisms of a nilpotent quotient of a free group. In the second part the authors apply these results to study the finite type invariants of \(\mathbb{Z} HS\), originally introduced by Ohtsuki [loc. cit.] and they partially answer questions 1 and 2 from [S. Garoufalidis, ibid. 5, No. 4, 441-461 (1996; Zbl 0889.57015), see the preceding review]. They reprove, in a more algebraic context, Ohtsuki’s fundamental result which states that the space of type \(m\) invariants of \(\mathbb{Z} HS\) is finite dimensional for every \(m\). Their proof allows to show (Corollary 3.8) that the graded space of degree \(m\) invariants of \(\mathbb{Z} HS\) is zero-dimensional unless \(m\) is divisible by 3. This partially answers question 1 of [Garoufalidis, loc. cit.]. Furthermore, the authors study a map from knots (in \(S^3)\) to \(\mathbb{Z} HS\), and show that type \(5m+1\) invariants of \(\mathbb{Z} HS\) map to type \(4m\) invariants of knots, thus making progress towards question 2 of [the first author, loc. cit.].

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
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References:

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