Gaps in the Milnor-Moore spectral sequence and the Hilali conjecture

Abstract

In his study of Halperin's toral-rank conjecture, M. R. Hilali conjectured that for any simply connected rationally elliptic space $X$, one must have $dim{\pi }_{\ast }(X)\otimes \mathbb{Q}\le dim{H}^{\ast }(X,\mathbb{Q})$. Let $(\Lambda V,d)$ denote a Sullivan minimal model of $X$ and $d_k$ the first non-zero homogeneous part of the differential $d$. In this paper, we use spectral sequence arguments to prove that if $(\Lambda V,d_k)$ is elliptic, then, there is no gaps in the $E_{\infty }$ term of the Milnor-Moore spectral sequence of $X$. Consequently, we confirm the Hilali conjecture when $V=V^{odd}$ or else when $k\ge 3$ and $(\Lambda V,d_k)$ is elliptic.

arXiv comment9 pages