The (rational) homotopy-by-homology ratio of a simply-connected CW-complex \(X\) with finite dimensional rational homology is \[h(X) = \frac{\dim\left(\pi_*\left(X\right)\otimes \mathbb Q \right)}{\dim \left(H_*\left(X;\mathbb Q\right)\right)} \in \mathbb Q^{\geq 0} \cup \left\{\infty\right\}.\]
This ratio seems to have been first explicitly considered in [O. Nakamura and T. Yamaguchi, Kochi J. Math. 6, 9–28 (2011; Zbl 1247.55007), Question 4.4]. It is natural to ask: for a given class \(\mathcal F\) of spaces, how does \(h(X)\) distribute as \(X\) runs over \(\mathcal F\)? For instance, the Hilali conjecture can be restated as asserting that for \(X\in\mathcal F\), for \(\mathcal F\) the class of rationally elliptic spaces, \(0\leq h(X) \leq 1\).
The paper under review considers a fiber sequence \(F\hookrightarrow E \to B\) of simply-connected rationally elliptic spaces and studies the relation between \(h(F), h(B)\), \(h(E)\) and \(h(F\times B)\).
The first elementary observation is that if the cohomology of the total space is isomorphic to the tensor product of fiber and base as \(H^*(B;\mathbb Q)\)-modules (i.e., if the rational Serre spectral sequence collapses), then \(h(E)=h(F) + h(B)\). This happens, for example, for the trivial fibration. Outside those very special cases, almost anything can happen. Even for spherical fibrations, the authors give simple examples for which all sensible inequalities between \(h(E), h(F\times B)\) and \(h(F)+h(B)\) hold.
Based on the heuristics of their examples, the authors propose that the following inequality is plausible: \[\frac{1}{2}h(F\times B) \leq h(E) < h(F)+h(B) + \frac{1}{4}.\] Assuming that the spaces \(F,E,B\) above are formal, the inequality has been recently proven (among other results relevant to the topic of this paper) in [M. Amann, “Homology versus homotopy in fibrations and in limits”, Preprint, arXiv:2006.03390].