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Homotopy groups and periodic geodesics of closed 4-manifolds. (English) Zbl 1331.53064
In this paper the authors present a connection between topology and geometry of closed 4-manifolds (with finite fundamental groups) by answering two questions: (i) how fast do the homotopy groups grow? and (ii) how many periodic geodesics are there? Given a simply connected, closed 4-manifold, the authors prove that the homotopy groups of such a manifold are determined by its second Betti number and the ranks of the homotopy groups can be explicitly calculated. They show that, for a generic metric on such a smooth 4-manifold with second Betti number at least three, the number of geometrically distinct periodic geodesics of length at most \(l\) grows exponentially as a function of \(l\). For proving the results the authors use the Milnor-Moore theorem as a common point to merge relevant ideas arising from rational homotopy theory and Koszul duality of associative algebras. The paper also contains an appendix on stable homotopy groups.

MSC:
53C22 Geodesics in global differential geometry
55P62 Rational homotopy theory
55Q10 Stable homotopy groups
57N65 Algebraic topology of manifolds
55Q52 Homotopy groups of special spaces
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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