Talk:Smoothings of products (Ex)

Latest revision as of 19:53, 29 August 2013

Let $<wikitex>; Let $\alpha \colon X \to BO$ and $\beta \colon Y \to BO$ be the given smooth structures. Recall the definition of $$ \Psi_\alpha \colon Conc(X) \to [X, PL/O]. $$ Given a smooth structure $\xi \colon X \to BO$, consider $\Delta_X^* (\xi \oplus- \alpha) \colon X \to BO$. In the diagram $$ \xymatrix{ & PL/O\ar[d] \ X\ar[r]_{\Delta_X^*(\xi \oplus -\alpha)}\ar[ur]^{\psi_\alpha(\xi)} & BO\ar[d]\ & BPL},$$ the lift $\psi_\alpha(\xi) $ (which is unique up to homotopy) results from the fact that the composition $X \xrightarrow{\Delta_X^*(\xi \oplus -\alpha)} BO \to BPL$ is nullhomotopic ($ \xi $ and $\alpha$ are lifts of the same PL structure). Then $$ \Psi_\alpha(\xi) = [\psi_\alpha(\xi)].$$ The PL homeomorphisms $f \colon X \to M$ and $g \colon Y \to N$ induce a smooth structure on $X \times Y$ via the map $$\mu f \oplus \nu g \colon X \times Y \xrightarrow{f \times g} M \times N \xrightarrow{\mu \times \nu} BO \times BO \xrightarrow{\oplus} BO,$$ where $\mu $ and $\nu$ classify the stable tangent bundle of $M$ and $N$, respectively. We now have $$ \Delta_{X \times Y}^*(\mu f \oplus \nu g \oplus -(\alpha \oplus \beta)) \simeq (\Delta_X \times \Delta_Y)^*(\mu f \oplus -\alpha \oplus \nu g \oplus -\beta) \simeq \Delta_X^*( \mu f \oplus -\alpha) \oplus \Delta_Y^*(\nu g \oplus -\beta), $$ where we have used that $ \oplus $ is homotopy associative and homotopy commutative. Since $PL/O$ is the homotopy fibre of the infinite loop map $BO \to BPL$, a lift of this map is now given by $ \psi_\alpha(\mu f) \oplus \psi_\beta(\nu g)$ ( $ \oplus $ denoting now the induced $H$-space structure on $ PL/O $). Therefore, we get $$ \Psi_{\alpha \times \beta}( f \times g) = \Psi_\alpha (f) \oplus \Psi_\beta(g)  = [ X \times Y \xrightarrow{\psi_\alpha (\mu f) \times \psi_\beta(\nu g)} PL/O \times PL/O \xrightarrow{\oplus} PL/O ].$$</wikitex>\alpha \colon X \to BO$ and $\beta \colon Y \to BO$ be the given smooth structures.

Recall the definition of

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

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

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

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

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

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

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

We now have

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

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

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

Therefore, we get

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