Homotopy groups and periodic geodesics of closed 4-manifolds

TL;DR

This paper explores the homotopy groups of closed 4-manifolds and demonstrates that for generic metrics with certain properties, the number of periodic geodesics and Reeb orbits grows exponentially with length.

Contribution

It establishes a link between the topology of 4-manifolds and the growth rate of periodic geodesics and Reeb orbits, providing explicit calculations and new insights.

Findings

Homotopy groups are determined by the second Betti number.

Number of periodic geodesics grows exponentially with length.

Number of Reeb orbits grows exponentially with length.

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.