Cohomological cross product
= Cohomological cross product
{title2=$a\times b=\pi_X^*a\smile\pi_Y^*b$}
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.