Talk:Smoothings of products (Ex)

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

Recall the definition of

$\displaystyle \Psi_\alpha \colon Conc(X) \to [X, PL/O].$ $$\displaystyle \Psi_\alpha \colon Conc(X) \to [X, PL/O].$$

Given a smooth structure $\xi \colon X \to BO$$\xi \colon X \to BO$, consider $\Delta_X^* (\xi \oplus- \alpha) \colon X \to BO$$\Delta_X^* (\xi \oplus- \alpha) \colon X \to BO$. In the diagram

$\displaystyle \xymatrix{ & PL/O\ar[d] \\ X\ar[r]_{\Delta_X^*(\xi \oplus -\alpha)}\ar[ur]^{\psi_\alpha(\xi)} & BO\ar[d]\\ & BPL},$

the lift $\psi_\alpha(\xi)$$\psi_\alpha(\xi)$ (which is unique up to homotopy) results from the fact that the composition $X \xrightarrow{\Delta_X^*(\xi \oplus -\alpha)} BO \to BPL$$X \xrightarrow{\Delta_X^*(\xi \oplus -\alpha)} BO \to BPL$ is nullhomotopic ($\xi$$\xi$ and $\alpha$$\alpha$ are lifts of the same PL structure). Then

$\displaystyle \Psi_\alpha(\xi) = [\psi_\alpha(\xi)].$ $$\displaystyle \Psi_\alpha(\xi) = [\psi_\alpha(\xi)].$$

The PL homeomorphisms $f \colon X \to M$$f \colon X \to M$ and $g \colon Y \to N$$g \colon Y \to N$ induce a smooth structure on $X \times Y$$X \times Y$ via the map

$\displaystyle \mu f \oplus \nu g \colon X \times Y \xrightarrow{f \times g} M \times N \xrightarrow{\mu \times \nu} BO \times BO \xrightarrow{\oplus} BO,$ $$\displaystyle \mu f \oplus \nu g \colon X \times Y \xrightarrow{f \times g} M \times N \xrightarrow{\mu \times \nu} BO \times BO \xrightarrow{\oplus} BO,$$

where $\mu$$\mu$ and $\nu$$\nu$ classify the stable tangent bundle of $M$$M$ and $N$$N$, respectively.

We now have

$\displaystyle \Delta_{X \times Y}^*(\mu f \oplus \nu g \oplus -(\alpha \oplus \beta)) \simeq (\Delta_X \times \Delta_Y)^*(\mu f \oplus -\alpha \oplus \nu g \oplus -\beta) \simeq \Delta_X^*( \mu f \oplus -\alpha) \oplus \Delta_Y^*(\nu g \oplus -\beta),$ $$\displaystyle \Delta_{X \times Y}^*(\mu f \oplus \nu g \oplus -(\alpha \oplus \beta)) \simeq (\Delta_X \times \Delta_Y)^*(\mu f \oplus -\alpha \oplus \nu g \oplus -\beta) \simeq \Delta_X^*( \mu f \oplus -\alpha) \oplus \Delta_Y^*(\nu g \oplus -\beta),$$

where we have used that $\oplus$$\oplus$ is homotopy associative and homotopy commutative.

Since $PL/O$$PL/O$ is the homotopy fibre of the infinite loop map $BO \to BPL$$BO \to BPL$, a lift of this map is now given by $\psi_\alpha(\mu f) \oplus \psi_\beta(\nu g)$$\psi_\alpha(\mu f) \oplus \psi_\beta(\nu g)$ ( $\oplus$$\oplus$ denoting now the induced $H$$H$-space structure on $PL/O$$PL/O$).

Therefore, we get

$\displaystyle \Psi_{\alpha \times \beta}( f \times g) = \Psi_\alpha (f) \oplus \Psi_\beta(g)  = [ X \times Y \xrightarrow{\psi_\alpha (\mu f) \times \psi_\beta(\nu g)} PL/O \times PL/O \xrightarrow{\oplus} PL/O ].$ $$\displaystyle \Psi_{\alpha \times \beta}( f \times g) = \Psi_\alpha (f) \oplus \Psi_\beta(g)  = [ X \times Y \xrightarrow{\psi_\alpha (\mu f) \times \psi_\beta(\nu g)} PL/O \times PL/O \xrightarrow{\oplus} PL/O ].$$