The external cup product is defined to be the map
$$ H^{k}( X ; R) \times H^{l}(Y ; R) \overset{\times}{\to} H^{k+l}(X \times Y; R)$$
where $a \times b = p_1^*(a) \smile p_2^*(b)$ where $p_1$ and $p_2$ are projections of $X \times Y$ onto $X$ and $Y$
I can't really figure out this definition.
Unfortunately, the example Hatcher uses is pretty complicated and not helpful.
Consider the space $X \times Y = S^n \times S^m$.
How do you apply this definition to the space $H^n(S^n ;R) \times H^m(S^m ;R)$ ?
Is every element in $H^{n+m}(S^n \times S^m ; R)$ a cup product of elements in $H^n(S^n ;R)$ and $H^m(S^ ;R)$ ??
How do I relate elements of $H^{n+m}(S^n \times S^m ; R)$ back to $H^n(S^n ;R)$ and $H^m(S^ ;R)$ ??
