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.

Glue the outer circles of two closed annuli by the identity, and their inner circles by complex conjugation. The result is a Klein bottle. To see the gluing explicitly, let $r\in [1/2,1]$ and $z\in S^1$. Parametrize the first annulus by $(z,\tau )=(z,2r-1)$, with $0\le \tau \le 1$, and the second by $(z,\tau )=(z,3-2r)$, with $1\le \tau \le 2$. The outer-circle gluing identifies their common level $\tau =1$. The remaining gluing isThis is the mapping torus of reflection of the circle. The coordinate maps and their piecewise inverses agree on every identified boundary, so the universal property of the quotient topology gives inverse continuous maps between the two quotient descriptions. Gluing collars by the reflection supplies disc charts at the seam; thus it is a closed surface. Cutting the circle coordinate at one point gives a rectangle with one matching and one reversed edge pairing, the usual Klein-bottle model. Its Euler characteristic is zero and it is a nonorientable surface, so the classification theorem identifies it with the connected sum of two real projective planes.

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.