Braiding of differential forms and homotopy types

Abstract

Let k be an arbitrary commutative ring. We associate fonctorially to any simplicial set X a differential graded algebra W ̂ ∗(X) with a globally 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 ̂ ∗(X) as a braided differential graded algebra 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 recent results of M.A. Mandell [5,6].