%0 Journal Article
%T Gaps in the Milnor-Moore spectral sequence and the Hilali conjecture
%A Youssef Rami
%J Mathematics
%D 2015
%I arXiv
%X 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 $dim\pi _*(X)\otimes \mathbb{Q} \leq dimH^*(X,\mathbb{Q})$. 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 $k\geq 3$ and $(\Lambda V, d_k)$ is elliptic.
%U http://arxiv.org/abs/1502.04200v2