Approximating rational spaces with elliptic complexes and a conjecture of Anick

Barry Jessup and Aniceto Murillo-Mas

Vol. 181 (1997), No. 2, 269–280

DOI: 10.2140/pjm.1997.181.269

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 > dimS. A functorial version of this conjecture due to McGibbon is that for any n there exists an elliptic complex E_n and an n-equivalence S → E_n. In fact, this is equivalent to its Eckmann-Hilton dual, which we prove in the rational category for a small class of simply connected spaces. Moreover, we construct the n-equivalence in such a way that the homotopy fibre is, rationally, a product of a finite number of odd spheres.

Milestones

Received: 10 June 1995

Revised: 17 November 1996

Published: 1 December 1997