On Hilali’s conjecture related to Halperin’s

Abstract

In this paper, we focus on Hilali’s conjecture that, for any simply-connected elliptic CW-complex X, the total sum of the rational Betti numbers is at least as large as the total rank of its rational homotopy. We investigate this conjecture for coformal spaces and suggest some research directions to resolve it completely. Finally, we put up a bridge between the Hilali conjecture and that of Halperin: the toral rank conjecture and use it to establish the latter holds for all manifolds of dimension less than 16 and whose toral rank is equal to 4.