Verifying the Hilali conjecture up to formal dimension twenty (Q777241)

A simply-connected CW-complex \(X\) is \emph{rationally elliptic} if \[ \dim \pi_*(X)\otimes \mathbb Q < \infty\text{ and }\dim H^*(X;\mathbb Q) < \infty. \] In his 1990 thesis, M.R. Hilali posed the following conjecture, now a classical one in rational homotopy theory. The Hilali Conjecture: If \(X\) is a simply-connected rationally elliptic CW-complex, then \[ \dim \pi_*(X)\otimes \mathbb Q \leq \dim H^*(X;\mathbb Q). \] For many families of rationally elliptic spaces, the conjecture is known to be true. However, the general case is still an open problem at the time of writing these lines -- no counterexamples exist. The paper under review verifies that the Hilali Conjecture holds in any rationally elliptic space of formal dimension at most \(20\). The formal dimension of a space \(X\) with finite dimensional cohomology is the largest \(n\) such that \(H^n(X;\mathbb Q) \neq 0.\) The verification involves calculations with computer software. First, a theorem of \textit{J. B. Friedlander} and \textit{S. Halperin} [Invent. Math. 53, 117--133 (1979; Zbl 0396.55010)] characterizes exactly what sequences of integers \(\{n_i\}\) occur as non-trivial rational vector spaces \(\pi_{n_i}(X)\otimes \mathbb Q\), for \(X\) a rationally elliptic space, in terms of a purely arithmetic condition satisfied among the integers of the given family. A sequence of integers \(\{n_i\}\) realizing the rational homotopy groups of some elliptic space does not determine the rational homotopy type of such a space, though. Typically, there will be many distinct rational homotopy types for such a given family; these distinct types have to be classified ``by hand''. \textit{O. Nakamura} and \textit{T. Yamaguchi} [Kochi J. Math. 6, 9--28 (2011; Zbl 1247.55007)] developed some software that computes the sequences of integers satisfying the Friedlander-Halperin condition up to a given dimension, and they used it to verify the Hilali Conjecture for spaces of formal dimension up to \(16\). The authors of the present paper provide a series of ad hoc results and simplification arguments that, when incorporated to the mentioned software and combined with some other computations they do, allow them to tell apart the distinct rational homotopy types of elliptic spaces of formal dimension up to 20 and verify the Hilali Conjecture on them. Along the way, the authors simplify arguments and correct some statements in some of the previous papers in the literature.