We give a geometric slice-like characterization for the vanishing of Milnor
's link invariants by proving the k-slice conjecture. This conjecture state
s that a link L has vanishing Milnor p-invariants of length less than or eq
ual to 2k 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 qu
otients of class k. In the course of our proof, we compute the dimension le
ss than or equal to 3 homology groups of finitely generated free nilpotent
Lie rings and groups. We develop a new algorithm for constructing a weighte
d chain resolution for a nilpotent group with torsion free lower central se
ries quotients, and with the property that its associated graded complex is
the Koszul complex of the associated graded Lie ring. This give a new deri
vation of the May spectral sequence relating the group homology of the nilp
otent group to the Lie ring homology of its associated graded Lie ring. Fin
ally, we define Tt-invariants of "pictures" and use these to describe a gen
erating set of cocycles in the cohomology of the free nilpotent groups. Som
e sample computations follow. (C) 2001 Elsevier Science Ltd. All rights res
erved.