Tbilisi Mathematical Journal

The Hilali conjecture on product of spaces

Shoji Yokura

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The Hilali conjecture claims that a simply connected rationally elliptic space $X$ satisfies the inequality $\dim (\pi_*(X)\otimes \mathbb{Q} ) \leqq \dim H_*(X;\mathbb{Q} )$. In this paper we show that for any such space $X$ there exists a positive integer $n_0$ such that for any $n \geqq n_0$ the strict inequality $\dim (\pi_*(X^n)\otimes \mathbb{Q} ) \lt \dim H_*(X^n; \mathbb{Q} )$ holds, where $X^{n}$ is the product of $n$ copies of $X$.


This work is supported by JSPS KAKENHI Grant Numbers JP16H03936 and JP19K03468.


The author would like to thank Toshihiro Yamaguchi for useful comments.

Article information

Tbilisi Math. J., Volume 12, Issue 4 (2019), 123-129.

Received: 29 September 2019
Accepted: 25 October 2019
First available in Project Euclid: 3 January 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 55P62: Rational homotopy theory
Secondary: 55Q40: Homotopy groups of spheres 55N99: None of the above, but in this section

Hilali conjecture rational homotopy theory


Yokura, Shoji. The Hilali conjecture on product of spaces. Tbilisi Math. J. 12 (2019), no. 4, 123--129. doi:10.32513/tbilisi/1578020572. https://projecteuclid.org/euclid.tbilisi/1578020572

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