I've heard the Klein bottle described as a 4D Mobius strip. If that's the case, when you cut it in half you get two Mobius strips and not, instead two interlocked hyper-cylinders? Am I misunderstanding something?
[Math] Cutting a Klein bottle in half.
klein-bottlemobius-band
Related Solutions
Imagine the Möbius strip as the unit square, where $(0,x)$ is identified with $(1,1-x)$. The line $\{(x,\frac{1}{2}):x\in[0,1]\}$ is a circle as a subspace of the Möbius strip. Then the map $H:M\times[0,1]\to M$ given by $H((x,y),t)=(x,(1-t)y+\frac{t}{2})$ gives a strong deformation retract of $M$ to the circle.
Below is a picture of the Klein bottle cut into two Möbius strips (along the orange line), so you can see what's going on. To answer your last question: you can apply the above deformation retract to map one of those Möbius bands to the circle. In particular the orange line, which is your $A\cap B$, gets mapped to the circle by this deformation retract.
Let's directly glue two Möbius strips into a Klein bottle.
- Take a Möbius strip and make its bottom wider:
- Make the rear part of the band like a half of a bottle, also the front part of the band like a tube.
- Take another Möbius strip and repeat steps 1 & 2 for the new one. Then glue new strip with the old one along their boundaries. Now we get a Klein bottle.
Therefore, a Klein bottle is homeomorphic to the glued Möbius strips.
Best Answer
The Klein bottle surface isn't really a 4-dimensional version of a Mobius strip. If you're looking for an analogue of the Mobius strip one dimension higher, your best bet is a solid Klein bottle, i.e. the space obtained from a solid cylinder $D^2 \times[0,1]$ (or solid cube $[0,1]^3$) by gluing the top to the bottom via a reflection. This space can also be described as the product of a Mobius strip with an interval.
The solid Klein bottle is a non-orientable 3-manifold with boundary, and it's analogous to the Mobius strip in the sense that a 3-manifold is orientable if and only if it doesn't contain a solid Klein bottle. The boundary of the solid Klein bottle is the Klein bottle surface.
As you mention, if you cut a Klein bottle in half lengthwise, it is possible to obtain two Mobius strips. However, it is also possible to cut a Klein bottle in half lengthwise to obtain a single long orientable strip, i.e. a cylinder $S^1\times[0,1]$. Roughly speaking, this depends on which lengthwise direction you use for the cut. Similarly, if you cut a solid Klein bottle in half lengthwise, you can obtain either two solid Klein bottles or a single long solid torus, depending on the direction of the cut.