On the Rank of Maps Induced by Fibrations in Homotopy and Homology
Keywords:
elliptic spaces, Hilali conjecture, rational homotopy theory, Sullivan models, Yamaguchi-Yokura conjectureAbstract
Let $F \rightarrow E \xrightarrow{P} B$ be a fibration of simply connected elliptic CW-complexes. Denote $\operatorname{Im} \pi_*(p) \otimes \mathbb{Q}$ by
$$
\oplus_i \operatorname{Im}\left\{\pi_i(p) \otimes \mathbb{Q}: \pi_i(E) \otimes \mathbb{Q} \rightarrow \pi_i(B) \otimes \mathbb{Q}\right\}
$$
and $\operatorname{Im} H_*(p ; \mathbb{Q})$ by
$$
\oplus_i \operatorname{Im}\left\{H_i(p ; \mathbb{Q}): H_i(E ; \mathbb{Q}) \rightarrow H_i(B ; \mathbb{Q})\right\} .
$$
The topological aspect of this paper is centered around answering the question
$$
\text { Is rank } \pi_*(p) \otimes \mathbb{Q} \leq \operatorname{rank} H_*(p ; \mathbb{Q}) \text { ? }
$$
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.
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