Milnor link invariants and quantum 3-manifold invariants

Abstract

Let $\mathcal{Z}(M)$ be the 3-manifold invariant of Le, Murakami and Ohtsuki. We show that $\mathcal{Z}(M)=1+o(n)$ , where $o(n)$ denotes terms of degree $\ge n$ , if M is a homology 3-sphere obtained from $S^3$ by surgery on an n-component Brunnian link whose Milnor $\overline{\mu }$ -invariants of length $\le 2n$ vanish. We prove a realization theorem which is a partial converse to the above theorem.¶Using the Milnor filtration on links, we define a new bifiltration on the $\mathbb{Q}$ vector space with basis the set of oriented diffeomorphism classes of homology 3-spheres. This includes the Milnor level 2 filtration defined by Ohtsuki. We show that the Milnor level 2 and level 3 filtrations coincide after reindexing.