Milnor link invariants and quantum 3-manifold invariants

Abstract

Let $ {\cal Z}(M) $ be the 3-manifold invariant of Le, Murakami and Ohtsuki. We show that $ {\cal Z}(M) = 1 + o(n) $ , where $ o(n) $ denotes terms of degree $ \geq 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 $ \leq 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 $ \Bbb 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.