Combinatorial models  for iterated loop spaces 
By F. Cohen and Ran  Levi

The objective of this paper is to provide free simplicial  group models for 
the functors \Omega^nX and \Omega^n\Sigma^{n+k}X. The models are based on 
classical constructions in simplicial homotopy theory. Specifically, Milnor's 
functor F, Kan's loops group functor G and the Moore loop space construction 
Ware used to produce these models. The models are given in terms of free 
groups with specific generators and the formulas defining the simplicial 
operators are given. The utility of these models is that in them certain 
maps can be written explicitly in a relatively easy way. To illustrate this 
a null homotopy of the commutator map on a double loop space is given. A 
simplcial model for the mapping space from a Riemann surface to an arbitrary 
space is also given. 
  

Stage: Submitted, Preprint available