Braiding of differential forms and homotopy types

Let k be an arbitrary commutative ring. We associate fonctorially to any si mplicial set X a differential graded algebra (W) over cap*(X) with a global ly defined braiding, which is an improvement of a previous work [3,4]. If k = Z and with some mild finiteness conditions on X, we show that the quasi- isomorphisms class of (W) over cap*(X) as a braided differential graded alg ebra determines the p-adic homotopy type of X for all the prime numbers p, and also the rational homotopy type. As in [3,4]? the proof uses some recen t results of M.A. Mandell [5,6]. (C) 2000 Academie des sciences/Editions sc ientifiques et medicales Elsevier SAS.