On Finite Type 3-manifold Invariants Ii

The purpose of the present paper is, among other things, to relate the seemingly unrelated notions of surgical equivalence of links in S 3 ((Le1]) and the notion of nite type invariants of oriented integral homology 3-spheres, due to T. Ohtsuki Oh]. The paper consists of two parts. In the rst part we classify pure braids and string links modulo the relation of surgical equivalence. We prove that the group of surgical equivalence classes of pure braids is isomorphic to the corresponding group of string links (Theorem 2). We also give two alternative descriptions of the above mentioned group P S E (n) of surgical equivalence classes of n strand pure braids: one as a semidirect product of P S 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 we apply these results to study the nite type invariants of ZHS, originally introduced by Ohtsuki Oh] and partially answer questions 1 and 2 from Ga]. We reprove, in a more algebraic context, Ohtsuki's fundamental result which states that the space of type m invariants of ZHS is nite dimensional for every m. Our proof allows us to show (Corollary 3.8) that the graded space of degree m invariants of ZHS is zero dimensional unless m is divisible by 3. This partially answers question 1 of Ga]. Furthermore, we study a map from knots (in S 3) to ZHS, and show that type 5m + 1 invariants of ZHS map to type 4m invariants of knots, thus making progress towards question 2 of Ga].