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 $h(X)= \frac{\dim \left( \pi_\ast(X) \otimes \mathbb{Q} \right)}{\dim H_\ast(X; \mathbb{Q})}$. In this paper, we propose to evaluate $h(X)$ of the total space $X$ of a fibration $\xi: F \hookrightarrow X \to B$ of elliptic spaces by $h(F)$, $h(B)$, and $h(F \times B)$. A conjectural formula is
$$ \frac{1}{2} \cdot h(F \times B) \leqq h(X) < h(F) + h(B) + \frac{1}{4} .$$
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