Floer homology of cotangent bundles and the loop product

Alberto Abbondandolo and Matthias Schwarz

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Submission date: 05. May. 2008
Pages: 109
published in: Geometry and topology, 14 (2010) 3, p. 1569-1722 
DOI number (of the published article): 10.2140/gt.2010.14.1569
Bibtex
MSC-Numbers: 53D40, 57R58
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Abstract:
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.