Tbilisi Mathematical Journal

The Hilali conjecture on product of spaces

Shoji Yokura

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

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1578020572

Digital Object Identifier
doi:10.32513/tbilisi/1578020572

Mathematical Reviews number (MathSciNet)
MR4047580

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

Keywords
Hilali conjecture rational homotopy theory

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


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References

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