Cohomological cross product ($a\times b={\pi }_X^{\ast }a⌣{\pi }_Y^{\ast }b$)

The cross product takes classes on two factors to a class on their product. Its multiplication rule has the graded tensor product sign: $(a\times b)(c\times d)=(-1{)}^{|b||c|}(ac)\times (bd)$. The Künneth theorem makes these products an additive basis when the integral cohomology of the finite cell complexes is torsion-free.