On the Rank of Maps Induced by Fibrations in Homotopy and Homology - Chouingou, Saloua; Zaim, Abdelhadi (Topology Proceedings, 2023)
Let $F → E \xrightarrow{P} B$ be a fibration of simply connected elliptic CW-complexes. Denote $\operatorname{Im} π_*(p) \otimes \mathbb{Q}$ by $$ \oplus_i \operatorname{Im}\left\{π_i(p) \otimes \mathbb{Q}: π_i(E) \otimes \mathbb{Q} → π_i(B) \otimes \mathbb{Q}\right\} $$ and $\operatorname{Im} H_*(p ; \mathbb{Q})$ by $$ \oplus_i \operatorname{Im}\left\{H_i(p ; \mathbb{Q}): H_i(E ; \mathbb{Q}) → H_i(B ; \mathbb{Q})\right\} . $$ The topological aspect of this paper is centered around answering the question $$ \text { Is rank } π_*(p) \otimes \mathbb{Q} ≤ \operatorname{rank} H_*(p ; \mathbb{Q}) \text { ? } $$ We are able to prove that, in general, the response must be negative, but in this paper, we will prove the positive in certain reasonable cases.