Normal invariants of products (Ex)

Let $(f, b) \colon M \to X$$<wikitex>; Let $(f, b) \colon M \to X$ and $(g, c) \colon N \to Y$ be degree one normal maps of closed smooth manifolds $X$ and $Y$ with normal invariants $\eta_X(f, b) \in [X, G/O]$ and $\eta_Y(g, c) \in [Y, G/O]$. {{beginthm|Exercise}} Determine the normal invariant, $$\eta_{X \times Y}(f \times g, b \times c \colon M \times N \to X \times Y) \in [X \times Y, G/O],$$ in terms of $\eta_X(f, b)$ and $\eta_Y(g, c)$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]](f, b) \colon M \to X$ and $(g, c) \colon N \to Y$$(g, c) \colon N \to Y$ be degree one normal maps of closed smooth manifolds $X$$X$ and $Y$$Y$ with normal invariants $\eta_X(f, b) \in [X, G/O]$$\eta_X(f, b) \in [X, G/O]$ and $\eta_Y(g, c) \in [Y, G/O]$$\eta_Y(g, c) \in [Y, G/O]$.

Exercise 0.1.

Determine the normal invariant,

$\displaystyle \eta_{X \times Y}(f \times g, b \times c \colon M \times N \to X \times Y) \in [X \times Y, G/O],$ $$\displaystyle \eta_{X \times Y}(f \times g, b \times c \colon M \times N \to X \times Y) \in [X \times Y, G/O],$$

in terms of $\eta_X(f, b)$$\eta_X(f, b)$ and $\eta_Y(g, c)$$\eta_Y(g, c)$.