Cohomology cross product
= Cohomology cross product
{title2=$a\times b=p_1^*a\smile p_2^*b$}
For classes on $X$ and $Y$, apply the <induced maps on cohomology> of the coordinate projections and take their <cup product> on $X\times Y$. The result has degree $|a|+|b|$. On a <tensor product> of graded rings, the sign rule $(a\otimes b)(c\otimes d)=(-1)^{|b||c|}ac\otimes bd$ makes this a ring homomorphism.