ANICK’S CONJECTURE AND INFINITIES IN THE MINIMAL MODELS OF SULLIVAN1

Abstract. An elliptic space is one whose rational homotopy and rational cohomol-ogy are both finite dimensional. David Anick conjectured that any simply connected finite CW-complex S can be realized as the k-skeleton of some elliptic complex as long as k> dim S, or, equivalently, that any simply connected finite Postnikov piece S can be realized as the base of a fibration F → E → S where E is elliptic and F is k-connected, as long as k> dim S. This conjecture is only known in a few cases, and here we show that in particular if the Postnikov invariants of S are decomposable, then the Anick conjecture holds for S. We also relate this conjecture with other finiteness properties of rational spaces. §1. Introduction. A topological space is elliptic if its rational homotopy and ra-tional cohomology are both finite dimensional. This class includes many interesting and well-known spaces which enjoy important structural properties ([H1], [FHT]). Elliptic spaces are very special, as the generic space is not elliptic, even amongst those satisfying one of the finiteness conditions. However, Anick conjectured th