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It is not. See Theorem 1.4 of this paper by James.this paper by I.M. James (Trans. AMS 84 (1957), 545-558).
In particular, there exists no homotopy associative multiplication on $S^n$ unless $n=1$ or $n=3$.
It is not. See Theorem 1.4 of this paper by James.
It is not. See Theorem 1.4 of this paper by I.M. James (Trans. AMS 84 (1957), 545-558).