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Homotopie rationnelle et croissance du nombre de géodésiques fermées. (French) Zbl 0562.55011

Let X be a 1-connected finite complex and \(\beta_ i=\dim H^ i(X^{S^ 1};{\mathbb{Q}})\) the ith Betti numbers of the space of free loops on X. The author proves that for a large class of spaces the sequence \((\beta_ i)\) has a polynomial (resp. exponential) growth if \(\dim (\Pi (X)\otimes {\mathbb{Q}})<\infty\) (resp. dim \(\Pi\) (X)\(\otimes {\mathbb{Q}}=\infty)\). As proved by M. Gromov [J. Differ. Geom. 13, 303- 310 (1978; Zbl 0427.58010)] the sequence \(\beta_ i\) plays a crucial role in studying the number of closed geodesics with prescribed length.
Reviewer: J.C.Thomas

MSC:

55P62 Rational homotopy theory
53C22 Geodesics in global differential geometry
55P35 Loop spaces

Citations:

Zbl 0427.58010
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References:

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