Homotopy groups and periodic geodesics of closed 4-manifolds

Abstract

Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calculated. We show that for a generic metric on such a smooth four manifold with second Betti number at least three, the number of geometrically distinct periodic geodesics of length at most l grow exponentially as a function of l. The number of closed Reeb orbits of length at most l on the spherization of the cotangent bundle also grow exponentially for any Reeb flow.

24 pages; added a result on closed Reeb orbits

References in corpus (4)

• Positive topological entropy of Reeb flows on spherizations
• The index growth and multiplicity of closed geodesics
• The Free Loop Space Homology of $(n-1)$-connected $2n$-manifolds
• The Weinstein conjecture with multiplicities on spherizations

Cited by in corpus (2)

• The Free Loop Space Homology of $(n-1)$-connected $2n$-manifolds
• A Morse theoretic description of the Goresky-Hingston product