Attaching maps in the standard geodesics problem on $S^2$
Abbas Bahri
Unstable manifolds of critical points at infinity in the variational problems relating to periodic orbits of Reeb vector-fields in Contact Form Geometry are viewed in this paper as part of the attaching maps along which these variational problems attach themselves to natural generalizations that they have. The specific periodic orbit problem for the Reeb vector-field $\xi_0$ of the standard contact structure/form of $S^3$ is studied; the extended variational problem is the closed geodesics problem on $S^2$. The attaching maps are studied for low-dimensional (at most $4$) cells. Some circle and ''loop" actions on the loop space of $S^3$, that are lifts (via Hopf-fibration map) of the standard $S^1$-action on the free loop space of $S^2$, are also defined. ''Conjugacy" relations relating these actions are established.
keywords: Attaching maps Circle and Loop actions Critical points at Infinity Contact form Geometry Loop spaces.
Recent results in contact form geometry
Abbas Bahri
After briefly recalling the pseudo-holomorphic approach in contact form geometry and after sketching the ways with which this approach defines invariants, we introduce another approach, of more technical type, which starts with a variational problem on Legendrian curves. We show how this approach leads also to the definition of a homology.
Ideally, this homology would be generated by a part of the Morse complex of the variational problem which would involve only periodic orbits. Because of the lack of compactness, it has some additional part which we had characterized in an earlier work [5].
Taking a variant of this approach, we give here a much more restrictive characterization of this additional part which should allow to compute it precisely.
This should indicate that the lack of compactness, seen as creation of additional punctures in the pseudo-holomorphic approach, is much more limited than what would be theoretically allowed and leaves hope that it can be completely computed. The proof of all our claims will be published in [6].
keywords: critical points at infinity. Reeb vector-fields Contact structures Legendrian curves

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