Klein bottle

The Klein bottle is obtained from a square by identifying one pair of opposite sides with matching direction and the other pair with reversed direction.

A one-sided section loop and a two-sided fiber loop in the reflection mapping torus have mod-two self-intersections one and zero and meet once. Their intersection form is $Q$. In the evaluation-dual cohomology basis, the Poincare duality pairing has matrix $Q^{-1}$, giving $t^2=0$ and $x^2=tx\ne 0$.

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.

The wedge ${\mathbb{RP}}^2\vee S^1$ has the same additive integral cohomology as the Klein bottle: $\mathbb{Z}$ in degree zero, $\mathbb{Z}$ in degree one, and $\mathbb{Z}/2$ in degree two. Products of positive-degree classes vanish because the degree-one class comes from the circle summand, so its cohomology ring is isomorphic to that of the Klein bottle.

Forthe transformations with even $c$ form an index-two translation subgroup. Its quotient is a torus, giving the orientation double cover $T^2\to K$.

Let $Y$ be the mapping cylinder of the orientation double cover $T^2\to K$ with its free end omitted. If a Hausdorff space contains $Y$ as an open subset, van Kampen extends the orientation quotient of ${\pi }_1(K)$ to a surjection of the ambient fundamental group onto $C_2$. The ambient space is therefore not simply connected.

In a flat fundamental square, straight horizontal, vertical and rational-slope lines project to closed geodesics. The central horizontal glide axis is one-sided, a vertical line is two-sided, and the line $(2t,t)$ closes with one transverse self-intersection.