On Ratios of Homotopy and Homology Ranks of Fibrations
Keywords:
Betti number, elliptic space, Sullivan minimal modelAbstract
For a simply connected CW complex $X$, we let $h(X)= \frac{\operatorname{dim}\left(\pi_(X) \otimes \mathbb{Q}\right)}{\operatorname{dim} H_(X ; \mathbb{Q})}$. In this paper, we propose to evaluate $h(X)$ of the total space $X$ of a fibration $\xi: F \hookrightarrow X \rightarrow B$ of elliptic spaces by $h(F), h(B)$, and $h(F \times B)$. A conjectural formula is
$$
\frac{1}{2} \cdot h(F \times B) \leqq h(X)<h(F)+h(B)+\frac{1}{4} .
$$
References
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