Abstract
Given the cotangent bundle T*Q of a smooth manifold with its canonical symplectic structure, and a Hamiltonian function on T*Q which is fiberwise asymptotically quadratic, its well-defined Floer homology with the pair-of-pants ring structure is ring-isomorphic to the singular homology of the free loop space of Q endowed with its loop product. The analogous statement is true for the based loop space versions and the Pontrjagin product. This article gives an overview of the construction of this ring isomorphism which is based on Legendre duality and moduli spaces of flow trajectories of hybrid type, which are half Floer trajectories for the Hamiltonian problem and half Morse trajectories for the Lagrangian problem.
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ABBONDANDOLO, A., SCHWARZ, M. (2006). NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY. In: Biran, P., Cornea, O., Lalonde, F. (eds) Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology. NATO Science Series II: Mathematics, Physics and Chemistry, vol 217. Springer, Dordrecht. https://doi.org/10.1007/1-4020-4266-3_02
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DOI: https://doi.org/10.1007/1-4020-4266-3_02
Publisher Name: Springer, Dordrecht
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