Smoothings of products (Ex)

Latest revision as of 19:56, 29 August 2013

Let $<wikitex>; 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 $$ \Psi_\alpha \colon \textup{Conc}(M) \equiv [M, PL/O] \quad \text{and} \quad \textup{Conc}(N) \equiv [N, PL/O].$$ Suppose that $f \colon M \cong X$ and $g \colon N \cong Y$ are homeomorphism from smooth manifolds. {{beginthm|Exercise}} Determine $$ \Psi_{\alpha \times \beta}(f \times g \colon M \times N \to X \times Y) \in [X \times Y, PL/O]$$ 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 $X \times Y$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]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 $X \times Y$.

[edit] References