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

TL;DR

This paper investigates the Milnor-Moore spectral sequence and confirms the Hilali conjecture for certain classes of rationally elliptic spaces using spectral sequence techniques.

Contribution

It proves the absence of gaps in the spectral sequence under specific conditions and verifies the Hilali conjecture in new cases.

Findings

No gaps in the spectral sequence if $( ext{Lambda} V, d_k)$ is elliptic.

Confirms Hilali conjecture when $V=V^{odd}$ or $k extgreater=3$ with elliptic $( ext{Lambda} V, d_k)$.

Spectral sequence methods are effective in studying rational homotopy conjectures.

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 $dim\pi_*(X)\otimes \mathbb{Q} \leq dimH^*(X,\mathbb{Q})$. 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.