Anick's conjecture for spaces with decomposable Postnikov invariants

Abstract

An elliptic space is one whose rational homotopy and rational cohomology 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 $F\,{\to}\,E\,{\to}\,S$ where $E$ is elliptic and $F$ is $k$-connected, as long as the $k$ is larger than the dimension of any homotopy class of $S$. 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.

The Omnibus Conjecture---disproved

“…In the same article the following conjecture is presented and then shown to imply Anick's conjecture (see [4,Proposition 5,p. 6]).…”

“…In [4] this famous problem in rational homotopy theory (see [2, Chapter 39, Problem 3, p. 516]) was formulated as Conjecture 6.1 (Anick). Any simply connected finite CW complex S can be approximated arbitrarily closely on the right by an elliptic space.…”