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 primesP and any groupG, letG→L _PG be a universal homomorphism fromG into a group which is uniquely divisible by primes not inP, and denote also byL _P the prolongation of this functor to simplicial groups. We prove that, ifX is any connected simplicial set andJ is any free simplicial group which is a model for the loop space ΩX, then the classifying space is homotopy equivalent to the localization ofX atP. Thus, there is a map which is universal among maps fromX into spacesY for which the semidirect products π_κ(Y)⋊ π₁(Y) are uniquely divisible by primes not inP. This approach also yields a neat construction of fibrewise localization. Both authors are supported by DGES under grant PB97-0202, and the first-named author has a DGR fellowship with reference code 1997FI 00467.