Homotopy groups and periodic geodesics of closed 4-manifolds

Samik Basu; Somnath Basu

doi:10.1142/s0129167x15500597

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arXiv · 1303.3328

Homotopy groups and periodic geodesics of closed 4-manifolds

Samik Basu·Somnath Basu

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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.

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BibTeX RIS

Samik Basu, Somnath Basu. 2013-12-19. Homotopy groups and periodic geodesics of closed 4-manifolds. https://doi.org/10.1142/s0129167x15500597

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