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Can the higher order $A_{\infty}$ multiplications defined by Fukaya be made trivial(by perturbing gradient trees) when Morse cochain complex is isomorphic to Morse cohomology, in which case the cup ...

My question is about Floer theory via symplectic surgery of Minimal Model program for finding basis of cohomology. Motivation: Perelman for solving Thurston's Geometrization Conjecture used some sort ...