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 product is associative, e.g. $S^n, \mathbb{C}P^n$ with the standard Morse funcitons on them?
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1$\begingroup$ Typically one needs to choose more data (e.g. several Morse functions or some other perturbation data on Stasheff trees) to define these operations? Probably the answer to your question will depend on how you choose this data... $\endgroup$– Daniel PomerleanoCommented Apr 2, 2019 at 16:16
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$\begingroup$ We fix the Morse function and perturb the gradient flow trees, thx. $\endgroup$– ArunCommented Apr 4, 2019 at 8:33
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$\begingroup$ Just a note about the degree condition: If the coefficient of $m_k(x_1,\dotsc, x_k)$ at $x_{k+1}$ is non-zero, then the Morse indices satisfy $\mu(x_1) + \dotsb + \mu(x_k) - \mu(x_{k+1}) + k - 2 = 0$, which is the dimension-$0$-condition on the moduli space of Morse trees. Assuming that the Morse complex of $S^n$ is generated by one generator $S$ in degree $0$ and one generator $N$ in degree $n$, the only solution of this equation for $k\ge 3$ is for $k = n+2$ and $x_1 = \dots = x_k = S$, $x_{k+1}=N$. I would guess that this must be also zero for some geometric reasons. $\endgroup$– PavelCommented Jun 12, 2020 at 13:57
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