Cohomology cross product ($a\times b=p_1^{\ast }a⌣p_2^{\ast }b$)

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.