The Milnor degree of a 3-manifold

Abstract. The Milnor degree of a 3-manifold is an invariant that records the maximum simplicity, in terms of higher order linking, of any link in the 3-sphere that can be surgered to give the manifold. This invariant is investigated in the context of torsion linking forms, nilpotent quotients of the fundamental group, Massey products and quantum invariants, and the existence of 3-manifolds with any prescribed Milnor degree and first Betti number is established. Along the way, it is shown that the number Mrk of linearly independent Milnor in-variants of degree k, for r-component links in the 3-sphere whose lower degree invariants vanish, is positive except in the classically known cases (when r = 1, and when r = 2 with k = 2, 4 or 6). This paper initiates a study of the Milnor degree, a 3-manifold invariant introduced by the authors in [5]. The definition is recalled and motivated below. All 3-manifolds considered here will be closed, connected and oriented. Any such mani-fold can be constructed by integral surgery on a framed link L in the 3-sphere S3, writte