This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T * M of a compact orientable manifold M . The first result is a new L ∞ estimate for the solutions of the Floer equation, which allows to deal with a larger -and more natural -class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M , in the periodic case, or of the based loop space of M , in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian which is the Legendre transform of a Lagrangian on T M , and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W 1,2 free or based loops on M .

We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas-Sullivan loop product on the singular homology of the loop space of M . We also prove related results concerning the Floer homological interpretation of the Pontrjagin product and of the Serre fibration. The techniques include a Fredholm theory for Cauchy-Riemann operators with jumping Lagrangian boundary conditions of conormal type, and a new cobordism argument replacing the standard gluing technique. 53D40, 57R58; 55N45 IntroductionLet M be a closed manifold, and let H be a time-dependent smooth Hamiltonian on T M , the cotangent bundle of M . We assume that H is 1-periodic in time and grows asymptotically quadratically on each fiber. Generically, the corresponding Hamiltonian system (0-1) x 0 .t/ D X H .t; x.t// has a discrete set ᏼ.H / of 1-periodic orbits. The free abelian group F .H / generated by the elements in ᏼ.H /, graded by their Conley-Zehnder index, supports a chain complex, the Floer complex .F .H /; @/. The boundary operator @ is defined by an algebraic count of the maps u from the cylinder R T to T M , solving the Cauchy-Riemann type equation (0-2) @ s u.s; t/ C J.u.s; t// @ t u.s; t/ X H .t; u.s; t// D 0; 8.s; t/ 2 R T ;and converging to two 1-periodic orbits of (0-1) for s ! 1 and s ! C1. Here J is the almost-complex structure on T M induced by a Riemannian metric on M , and (0-2) can be seen as the negative L 2 -gradient equation for the Hamiltonian action functional.This construction is due to A Floer (eg [21;22;23;24]) in the case of a closed symplectic manifold P , in order to prove a conjecture of Arnold on the number of Since the space W 1;2 .T ; M / is homotopy equivalent to ƒ.M /, we get the required isomorphismGeometry & Topology, Volume 14 (2010) Floer homology of cotangent bundles and the loop product 1571 from the singular homology of the free loop space of M to the Floer homology of T M .Additional interesting algebraic structures on the Floer homology of a symplectic manifold are obtained by considering other Riemann surfaces than the cylinder as domain for the Cauchy-Riemann type equation (0-2). By considering the pair-of-pants surface, a noncompact Riemann surface with three cylindrical ends, one obtains the pairof-pants product in Floer homology (see the second author's thesis [46] and McDuff and Salamon [38]). When the symplectic manifold P is closed and symplectically aspherical, this product corresponds to the standard cup product from topology, after identifying the Floer homology of P with its singular cohomology by Poincaré duality, while when the manifold P can carry J -holomorphic spheres, the pair-of-pants product corresponds to the quantum cup product of P (see Piunikhin, Salamon and Schwarz [39] and Liu and Tian [37]).The main result of this paper is that in the case of cotangent bundles, the pair-of-pants product is also equivalent to a product on H .ƒ.M // coming from topology, but a more interesting one than the simple cup produ...

We study the action functional associated to a smooth Lagrangian function on the tangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H 1 curves, it is a Lyapunov function for some smooth Morse-Smale vector field, under the generic assumption that all the critical points are non-degenerate. This fact is sufficient to associate a Morse complex to the Lagrangian action functional.

The systolic ratio of a contact form α on the three-sphere is the quantitywhere T min (α) is the minimal period of closed Reeb orbits on (S 3 , α). A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that ρ sys ≤ 1 in a neighbourhood of the space of Zoll contact forms on S 3 , with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a local middle-dimensional non-squeezing theorem, and a sharp systolic inequality for Finsler metrics on the two-sphere which are close to Zoll ones. Our second main result is that ρ sys is unbounded from above on the space of tight contact forms on S 3 .

No abstract