Volume 12, issue 4 (2012)

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Local Floer homology and infinitely many simple Reeb orbits

Mark McLean

Algebraic & Geometric Topology 12 (2012) 1901–1923
Abstract

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.

Keywords
symplectic homology, local Floer, cotangent bundle, Reeb orbits
Mathematical Subject Classification 2010
Primary: 53D10, 53D25, 53D40