Commentarii Mathematici Helvetici

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Volume 74, Issue 2, 1999, pp. 322–344
DOI: 10.1007/s000140050092

Published online: 1999-06-30

Milnor link invariants and quantum 3-manifold invariants

N. Habegger[1] and Kent E. Orr[2]

(1) Université de Nantes, France
(2) Indiana University, Bloomington, USA

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.

Keywords: Kontsevich Integral, Milnor link invariants, 3-manifolds Homology spheres, quantum invariants, Feynman diagrams, finite type invariants

Habegger N., Orr Kent: Milnor link invariants and quantum 3-manifold invariants. Comment. Math. Helv. 74 (1999), 322-344. doi: 10.1007/s000140050092