Integral cohomology ring of the Klein bottle
= Integral cohomology ring of the Klein bottle
{c}
{title2=$H^*(K;\mathbb Z)$}
The integral cohomology groups of the Klein bottle are $H^0(K;\mathbb Z)=\mathbb Z$, $H^1(K;\mathbb Z)=\mathbb Z$, $H^2(K;\mathbb Z)=\mathbb Z/2$, and zero in higher degrees. The degree-one generator is pulled back from the base circle of the circle-bundle presentation, so its square is zero; hence every product of positive-degree classes vanishes.