Milnor link invariants and quantum 3-manifold invariants

  • N. Habegger

    Université de Nantes, France
  • Kent E. Orr

    Indiana University, Bloomington, USA


Let Z(M){\cal Z}(M) be the 3-manifold invariant of Le, Murakami and Ohtsuki. We show that Z(M)=1+o(n){\cal Z}(M) = 1 + o(n) , where o(n)o(n) denotes terms of degree n\geq n , if M is a homology 3-sphere obtained from S3S^3 by surgery on an n-component Brunnian link whose Milnor μ\overline\mu -invariants of length 2n\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 Q\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.

Cite this article

N. Habegger, Kent E. Orr, Milnor link invariants and quantum 3-manifold invariants. Comment. Math. Helv. 74 (1999), no. 2, pp. 322–344

DOI 10.1007/S000140050092