Does Milnor K-Theory arise from Waldhausen K-Theory Quillens higher K-groups of rings can be realized as πnK(C) - the Waldhausen K-Theory of a suitable Waldhausen category C. Is this also true for Milnor K-Theory of Rings? Is there a functor F from rings to waldhausen categories s.t. $K^M_n(R)\cong \pi_n(K(F(R))$?
 A: I don't know if there any evidence for this to be true. Note that Quillen K-groups are defined as homotopy groups of some space (+-construction, Q-construction, Waldhausen construction etc), whereas Milnor K-groups were defined in terms of generators and relations,
which generalize generators and relations for classical K_2.
More invariantly Milnor K-groups can be constructed using homology of GL_n (paper of Suslin and Nesterenko) or as certain motivic cohomology groups of a field (Suslin-Voevodsky).
However, these constructions are unrelated to any homotopy groups.
Also, I'm not sure how you define Milnor K-theory for a general ring R?
(I was interpreting your question with "ring R" replaced by "field F".)
A: Bob Thomason proved that there is no Milnor K-theory functor for schemes,
with a reasonable map to Quillen K-theory, in:

Le principe de scindage et
  l'inexistence d'une $K$-theorie de
  Milnor globale. [The splitting
  principle and the nonexistence of a
  global Milnor $K$-theory] Topology 31
  (1992), no. 3, 571--588.



*

*John

A: The original question seems not to have been answered yet.  One answer might be that it would be unnatural to expect all the Milnor K-groups of a field R to arise as the homotopy groups of a single space $K(F(R))$, because the natural way they currently arise is as homotopy groups of separate spaces, or better, of separate spectra.  The spectra are the Eilenberg-MacLane spectra $\mathbb Z(n)$ associated to the chain complexes that compute motivic cohomology of $R$, namely, $K_n^M R = \pi_{-n} \mathbb Z(n)$.
