I'm asking this question purely out of curiosity.

Let $\{M_\alpha\}_{\alpha\in A}$ be a collection of closed smooth manifolds, with exactly one in every diffeomorphism class of closed smooth manifold. Hence for each $\alpha,\beta\in A$, there exists a unique $\alpha\ast\beta\in A$ for which $M_{\alpha\ast\beta}$ is diffeomorphic to $M_\alpha\times M_\beta$. Hence we get an binary operation $\ast:A\times A\to A$ which is associative and commutative.

Can we choose diffeomorphisms $\phi_{\alpha\beta}:M_\alpha\times M_\beta\to M_{\alpha\ast\beta}$ which are associative? what about associative and commutative?

To be precise, associativity means the following diagram commutes: $$\begin{matrix}M_\alpha\times M_\beta\times M_\gamma&\rightarrow&M_{\alpha\ast\beta}\times M_\gamma\cr\downarrow&&\downarrow\cr M_\alpha\times M_{\beta\ast\gamma}&\rightarrow&M_{\alpha\ast\beta\ast\gamma}\end{matrix}$$ and commutativity means the following diagram commutes: $$\begin{matrix}M_\alpha\times M_\beta&\rightarrow&M_{\alpha\ast\beta}\cr\downarrow&&\downarrow\cr M_\beta\times M_\alpha&\rightarrow&M_{\beta\ast\alpha}\end{matrix}$$

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