Rational homotopy of the space of sections of a nilpotent bundle

Abstract

We show that an algebraic construction proposed by Sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle. In his paper Uhomologie des espaces fonctionnels, R. Thom studied the homotopy type of the space F* of continuous maps of X into F homotopic to a given map/. Starting from a Postnikov decomposition of F, he built the functional space F* step by step. He also indicated how one could construct a differential graded algebra describing the rational homotopy type of F*. Later on, Sullivan gave an algebraic model which mirrors this construction in terms of a DG-algebra representing X and the minimal model of F. The aim of this paper is to show, following the method of Thom, that the model of Sullivan is indeed a model for the functional space under suitable restrictions. As in (3), we consider the slightly more general problem of the determination of the rational homotopy type of the space of sections Ts of a nilpotent fiber space p: Y -» X homotopic to a given section s. In §1 we explain Thorn's geometric construction. In §2 we describe an algebraic model for an abelian Galois covering of a nilpotent space. In §3 we show how the model of Sullivan fits with the geometry. I thank the referee for many improvements of this paper. 1. A Postnikov factorization of the space of sections. 1.1. Let G be a finitely generated abehan group and let X be a path connected space whose cohomology groups Hk(X G) are finitely generated for each k. To avoid difficulties with the topologies (cf. (4)), we can work in the category of simplicial sets. Proposition (Thom (4)). The space K(G, m)x of continuous maps of X in the Eilenberg-Mac Lane complex K(G, m) is homotopically equivalent to the product nr=rA °fthe Eilenberg-Mac Lane spaces Ki = K(Hm-'(X; G), i). More precisely, let x G Hm(K(G; m); G) be the fundamental class of K(G; m). If e:K(G,m)X XX^K(G,m)