A note on the Hilali conjecture

TL;DR

This paper verifies the Hilali conjecture for two-stage spaces and a specific class of fibrations, showing the rational cohomology dimension exceeds or equals the rational homotopy groups, advancing understanding in rational homotopy theory.

Contribution

It proves the Hilali conjecture for two-stage spaces and a new class of spaces related to fibrations, expanding the conjecture's verified cases.

Findings

Hilali conjecture holds for two-stage spaces

Hilali conjecture verified for a class of fibrations

Rational cohomology dimension ≥ rational homotopy groups

Abstract

In this short note we observe that the Hilali conjecture holds for two-stage spaces, i.e. we argue that the dimension of the rational cohomology is at least as large as the dimension of the rational homotopy groups for these spaces. We also prove the Hilali conjecture for a class of spaces which puts it into the context of fibrations.