The general context of this paper is the following open problem in the algebraic aspects of rational homotopy theory, due to Hilali: given a rationally elliptic space, i.e. a simply connected space whose total rational homotopy and total (unreduced) rational cohomology are both finite-dimensional, is the dimension of the former less than or equal to the dimension of the latter? That is, denoting \(h(X) = \frac{\mathrm{dim} \, \pi_*(X) \otimes \mathbb{Q}}{ \mathrm{dim} \, H^*(X;\mathbb{Q})}\) for a rationally elliptic space, the question is whether we necessarily have \(0 \leq h(X) \leq 1\). The extremes are realized e.g. by a point and the two-sphere, respectively. Interestingly, as pointed out by O. Nakamura and T. Yamaguchi [Kochi J. Math. 6, 9–28 (2011; Zbl 1247.55007)], there are no known examples of elliptic spaces where \(h(X) \in (\frac{5}{6}, 1)\). (The value \(\frac{5}{6}\) is achieved by the total space of the pullback of the quaternionic Hopf fibration \(S^7 \to S^4\) via a non-zero degree map \(S^2 \times S^2 \to S^4\).)
Consider now (rational) fibrations of rationally elliptic spaces \(F \to X \to B\). (Note that if all three spaces are simply connected, then any two of them being rationally elliptic implies the third is also rationally elliptic).
T. Yamaguchi and S. Yokura [Topol. Proc. 58, 85–92 (2021; Zbl 1448.55015)] conjectured that \[ \frac{1}{2} h(F \times B) \leq h(X) < h(F) + h(B) + \frac{1}{4}. \] The author proves that this conjecture holds if \(X\) furthermore has positive Euler characteristic, or if \(F\) has positive Euler characteristic and the inclusion of the fiber \(F \to X\) is rationally cohomologically surjective. More generally, it is proved that if \(F\) has positive Euler characteristic (the Euler characteristic of a rationally elliptic space is \(\geq 0\), so \(X\) having positive Euler characteristic is a special case of this), then the left-hand inequality of the conjecture holds, i.e. \(\frac{1}{2} h(F \times B) \leq h(X)\).
The author also proves that if \(F,X,B\) are formal and rationally elliptic, then the conjecture holds. One should keep in mind that a rationally elliptic space with positive Euler characteristic is formal, so this is the scenario we find ourselves in if \(\chi(X) > 0\). It is also shown that the conjecture holds whenever \(H^*(X;\mathbb{Q})\) has at most one even and at most one odd generator as a \(\mathbb{Q}\)-algebra; in particular it holds if \(X\) is rationally homotopy equivalent to a simply connected closed homogeneous space of positive sectional curvature.
More generally, a somewhat weaker version of the left-hand inequality of the conjecture is proven to hold. Namely, for any fibration of rationally elliptic spaces \(F \to X \to B\), we have \[ \frac{1}{3}h(F\times B) \leq h(X). \] The author then goes on to consider notions of convergence for families \(\mathcal{X}\) of rationally elliptic spaces in terms of the function \(h\, \colon \mathcal{X} \to \mathbb{R}_{\geq 0} \cup \{+\infty\}\) to the extended reals.
We say the family converges to \(c\) if there are infinitely many \(X \in \mathcal{X}\) with \(h(X)\) landing in an arbitrarily small neighborhood of \(c\) (i.e. \(c\) is an accumulation point), and if \(c\) is the only real number with this property. For example, the family of complex projective spaces converges to \(0\); the family of all spheres does not converge since both \(\frac{1}{2}\) and \(1\) are accumulation points (achieved by odd and even spheres, respectively).
Another notion of convergence, meant to better capture families with arbitrarily large rational homotopy, is the following: we say \(\mathcal{X}\) \(\pi\)-converges to \(c\) if arbitrary neighborhoods of \(c\) contain \(h(X)\) for some \(X\) with arbitrarily large \(\dim \pi_* \otimes \mathbb{Q}\). For example, the family of all finite products of all even-dimensional spheres has accumulation points \(\{ 0 \} \cup \{\frac{2k}{2^k}\}_{k \geq 1}\), but only \(0\) is an accumulation point for \(\pi\)-convergence.
It is shown that in the family of elliptic spaces, any value \(h(X)\) achieved by a pure (examples of pure spaces are those elliptic spaces with positive Euler characteristic) or two-stage space is achieved by infinitely many spaces of the same type. Furthermore, the family of two-stage spaces \(\pi\)-converges to 0.