The Milnor degree of a three-manifold

TL;DR

This paper introduces the Milnor degree as a new invariant for 3-manifolds, exploring its properties and relationships with other invariants, and demonstrates the existence of manifolds with arbitrary Milnor degrees and Betti numbers.

Contribution

It defines the Milnor degree invariant for 3-manifolds and investigates its connections with classical invariants, establishing existence results for manifolds with prescribed invariants.

Findings

The Milnor degree relates to torsion linking forms, Massey products, and quantum invariants.

Existence of 3-manifolds with any given Milnor degree and first Betti number is proven.

The number of independent Milnor invariants of a certain degree is positive except in known special cases.

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 M(k,r) of linearly independent Milnor invariants of degree k, distinguishing 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).