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Floer homology of cotangent bundles and the loop product
Alberto Abbondandolo and Matthias Schwarz
In a previous paper we constructed an isomorphism between Floer homology on the cotangent bundle of an oriented, closed manifold $M$ for a suitable class of Hamiltonians of fibrewise quadratic type, and the homology of the free loop space of $M$. In this paper, we show that this isomorphism is also a ring isomorphism if we endow Flow homology with its natural pair-of-pants ring structure and the free loop space homology with the Chas-Sullivan loop product.
In particular, we show that also the pair-of-pants product factors through a variant of Floer homology associated to figure-8-loops, compatible with the factorization of the Chas-Sullivan loop product. This makes repeated use of Lagrangian boundary conditions of conormal type.