A homotopy idempotent construction by means of simplicial groups

We obtain a simplicial group model for localization of (not necessarily nilpotent) spaces at sets of primes by applying a suitable functor dimensionwise, as in earlier work of Quillen and Bousfield-Kan. For a set of primes P and any group G, let G → LPG be a universal homomorphism from G into a group which is uniquely divisible by primes not in P, and denote also by LP the prolongation of this functor to simplicial groups. We prove that, if X is any connected simplicial set and J is any free simplicial group which is a model for the loop space ΩX, then the classifying space W̄LPJ is homotopy equivalent to the localization of X at P. Thus, there is a map X → W̄LPJ which is universal among maps from X into spaces Y for which the semidirect products πk(Y) ⋊ π1(Y) are uniquely divisible by primes not in P. This approach also yields a neat construction of fibrewise localization.