Cutting a Klein bottle in half.

Question

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?

Answer 1

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.