Milnor link invariants and quantum 3-manifold invariants
The authors define a bifiltration on the \(\mathbb Q\) vector space generated by oriented diffeomorphism classes of integral homology spheres. As usual let \([M,L]\) denote the alternating sum over all sublinks, of the manifold obtained by surgery on the sublink. The new bifiltration is, \(\mathcal S^r_n\), the subspace of \(\mathcal S\) generated by admissible \(n\) component links with vanishing Milnor \(\overline {\mu}\)-invariants of length \(\leq r\). The filtration, \(\mathcal S^0_n\), is the universal LMO filtration. An invariant which vanishes on \(\mathcal S^0_n\) for some value of \(n\) is called a finite type 3-manifold invariant. There is a universal finite type 3-manifold invariant, \(Z(M)\), with values in the algebra of all Feynman diagrams. Any finite type 3-manifold invariant factors through \(Z(M)\). The main vanishing result in this paper is that \(\mathcal S_n^r \leq \mathcal S^0_n - [\frac{2n}{(r+1)}]\). This paper also includes a formula for the finite type invariant of the closure of a string link in terms of the true like part of the finite type invariant of the string link.
scientific article; zbMATH DE number 1333326