Smoothings of products (Ex)

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Let X and Y be closed PL n-manifolds, n \geq 5, and let X_\alpha and Y_\beta be smooth structures on X and Y respectively defining bijections

\displaystyle  \Psi_\alpha \colon \textup{Conc}(X) \equiv [X, PL/O] \quad \text{and} \quad \textup{Conc}(Y) \equiv [Y, PL/O].

Suppose that f \colon M \cong X and g \colon N \cong Y are homeomorphisms from smooth manifolds.

Exercise 0.1.

Determine
\displaystyle  \Psi_{\alpha \times \beta}(f \times g \colon M \times N \to X \times Y) \in [X \times Y, PL/O]

in terms of \Psi_\alpha(f \colon M \to X) and \Psi_\beta(g \colon N \to Y). Here \Psi_{\alpha \times \beta} is defined using the smooth structure X_\alpha \times Y_\beta on the PL manifold X \times Y.

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