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 are 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 grows exponentially as a function of l. The number of closed Reeb orbits of length at most l on the spherization of T * M also grow exponentially for any Reeb flow.

Homotopy groups of highly connected manifolds

“…It is enough to compute the dimension l d of the d th -graded part of the Lie algebra L u r (M ). We use the generating series to compute this from the universal enveloping algebra H * (ΩM ) as in [5].…”

“…Curiously, if r ≥ 1, one may construct a circle bundle over M whose total space is a connected sum of (r − 1) copies of S 2 × S 3 (cf. [5,13]). It follows that the homotopy groups, upto isomorphism, depend only on the integer r and not on the attaching map.…”

“…Theorem 4.5). From this expression the results in [5] about the rational homotopy groups may also be deduced.…”

“…These techniques are used to compute the homology of the loop space of the manifold using the cobar construction on the homology of M . A naive version of these ideas was used in an earlier paper [5].…”

Free loop space homology of highly connected manifolds

“…Free loop space homology of simply connected closed 4-manifolds has been studied in [BB13], but the methods used there do not extend to higher dimensions. Theorem 1.3 generalizes [BB13, Theorem C(1)].…”