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.