Local Floer homology and infinitely many simple Reeb orbits

Let $Q$ be a Riemannian manifold such that the Betti numbers of its free loop space with respect to some coefficient field are unbounded. We show that every contact form on its unit cotangent bundle supporting the natural contact structure has infinitely many simple Reeb orbits. This is an extension of a theorem by Gromoll and Meyer. We also show that if a compact manifold admits a Stein fillable contact structure then there is a possibly different such structure which also has infinitely many simple Reeb orbits for every supporting contact form. We use local Floer homology along with symplectic homology to prove these facts.

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Keywords

symplectic homology, local Floer, cotangent bundle, Reeb orbits

Mathematical Subject Classification 2010

Primary: 53D10, 53D25, 53D40

References

Publication

Received: 8 February 2012

Revised: 19 June 2012

Accepted: 21 June 2012

Published: 17 September 2012

Authors

Mark McLean
Department of Mathematics MIT Building 2, Room 275 77 Massachusetts Avenue Cambridge, MA 02139-4307 USA
http://math.mit.edu/~mclean/

Volume 12, issue 4 (2012)

Mark McLean

Abstract