On Ratios of Homotopy and Homology Ranks of Fibrations - Yamaguchi, Toshihiro; Yokura, Shoji (Topology Proceedings, 2020)
For a simply connected CW complex $X$, we let $h(X)= \frac{\dim \left( π_*(X) \otimes \mathbb{Q} \right)}{\dim H_*(X; \mathbb{Q})}$. In this paper, we propose to evaluate $h(X)$ of the total space $X$ of a fibration $ξ: F \hookrightarrow X → B$ of elliptic spaces by $h(F)$, $h(B)$, and $h(F × B)$. A conjectural formula is $$ \frac{1}{2} · h(F × B) \leqq h(X) < h(F) + h(B) + \frac{1}{4} .$$