## Tbilisi Mathematical Journal

### The Hilali conjecture on product of spaces

Shoji Yokura

#### Abstract

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 $X$.

#### Note

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

#### Note

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

#### Article information

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

Dates
Accepted: 25 October 2019
First available in Project Euclid: 3 January 2020

https://projecteuclid.org/euclid.tbilisi/1578020572

Digital Object Identifier
doi:10.32513/tbilisi/1578020572

Mathematical Reviews number (MathSciNet)
MR4047580

#### Citation

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

#### References

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• T. Yamaguchi and S. Yokura, On ratios of homotopy and homology ranks of fibrations, preprint, May 2019.