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Title: 2. Relations among Morse A_\infty-Category, De Rham A_\infty-Category and Fukaya Category
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Given a compact manifold M with a suitable metric, we can build its Morse A_\infty-Category, De Rham A_\infty-Category and the Fukaya Category of its cotan- gent bundle. In this talk, I will show that the Morse A_\infty-Category is quasi- isomorphic to the De Rham A_\infty-Category and the Morse A_\infty-Category can be embedded to the Fukaya Category of the cotangent bundle. The results may not be that interesting since the De Rham A_\infty-Category is a trivial A_\infty-Category, but the tools we use for the proof are somehow interesting and useful. The quasi-isomorphism between the Morse A_\infty-Category and the De Rham A_\infty-Category can be proved by using the homological pertubation theory, established by Kontsevich and Soibelman. And the construction of the projector in the homological pertubation theory uses the finite flow method given by Harvey and Lawson. The embedding of Morse A_\infty-Category to the Fukaya category can be given by identifying the moduli space of gradient trees and the moduli space of pseudo-holomorphic discs in the cotangent bundle, which is established by Fukaya and Oh. And for the identification we need the implicit function theorem given by McDuff and Salamon. In the talk I will explain these ideas and implement these strategies.