Combinatorial models for iterated loop spaces By: F. R. Cohen and R. Levi AMS Classification: 55R35 Address: Fred Cohen, Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A. Ran Levi, Department of Mathematical Sciences, University of Aberdeen, Aberdeen, AB24 3UE, Scotland Email: Fred Cohen Ran Levi The objective of this paper is to provide free simplicial group models for the functors $\Omega^n X$ 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 $\Omega$ are 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. Similar ideas are used to give a model for pointed mapping spaces out of a Riemann surface.