Méthodes quantiques en topologie algébrique

Abstract

Using quantum methods, we introduce here the notion of “neo-algebra” which generalizes the notion of a commutative differential graded algebra. Under some mild finiteness conditions, we can associate functorially to a space a neo-algebra over the finite field F p : its quasi-isomorphism's class determines the p-adic-homotopy type of X. As a matter of fact, from this data, we can describe in a simple way Steenrod operations in the cohomology of X, as well as the p primary part of its homotopy groups. This point of view extends to finite characteristics the well-known rational homotopy theory of D. Quillen [9] and D. Sullivan [11]. It is deeply related to previous works of P. May, I. Kriz [5] and M.A. Mandell [7], [8].