Resonance for loop homology of spheres
| Autor: | Hingston, Nancy, Rademacher, Hans-Bert |
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| Rok vydání: | 2011 |
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| Druh dokumentu: | Working Paper |
| Popis: | A Riemannian or Finsler metric on a compact manifold M gives rise to a length function on the free loop space \Lambda M, whose critical points are the closed geodesics in the given metric. If X is a homology class on \Lambda M, the minimax critical level cr(X) is a critical value. Let M be a sphere of dimension >2, and fix a metric g and a coefficient field G. We prove that the limit as deg(X) goes to infinity of cr(X)/deg(X) exists. We call this limit the "global mean frequency" of M. As a consequence we derive resonance statements for closed geodesics on spheres; in particular either all homology on \Lambda M of sufficiently high degreee lies hanging on closed geodesics whose mean frequency (average index / length) equals the global mean frequency, or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to the global mean frequency. The proof uses the Chas-Sullivan product and results of Goresky-Hingston [GH]. Comment: Revision of the paper following suggestions of the referee |
| Databáze: | arXiv |
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