Links, pictures and the homology of nilpotent groups

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We give a geometric slice-like characterization for the vanishing of Milnor's link invariants by proving the k-slice conjecture. This conjecture states that a link L has vanishing Milnor μ ̄ -invariants of length ⩽2 k if and only if L bounds disjoint surfaces in a four disk in such a way that the fundamental group of the complement admits free nilpotent quotients of class k. In the course of our proof, we compute the dimension ⩽3 homology groups of finitely generated free nilpotent Lie rings and groups. We develop a new algorithm for constructing a weighted chain resolution for a nilpotent group with torsion free lower central series quotients, and with the property that its associated graded complex is the Koszul complex of the associated graded Lie ring. This give a new derivation of the May spectral sequence relating the group homology of the nilpotent group to the Lie ring homology of its associated graded Lie ring. Finally, we define μ ̄ -invariants of “pictures” and use these to describe a generating set of cocycles in the cohomology of the free nilpotent groups. Some sample computations follow.