Smoothings of products (Ex)

Let $X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X$ and $Y$$Y$ be closed PL $n$$n$-manifolds, $n \geq 5$$n \geq 5$, and let $X_\alpha$$X_\alpha$ and $Y_\beta$$Y_\beta$ be smooth structures on $X$$X$ and $Y$$Y$ respectively defining bijections

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

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

Exercise 0.1.

Determine

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

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