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A question was asked in my topology course the other day (not an assignment for credit).

Let $f:S^1\rightarrow S^1$ by $f(z)=z^2$ ($S^1$ is considered to be in the complex plane). What is the mapping cylinder of $f$?

After discussing it briefly with a few others, I was told it was actually the Möbius band. But this is very difficult to visualize. For example, the Möbius band has only one edge, but a mapping cylinder has two, the domain at the "top" and the bottom slice which is glued to the image of $f$. The image I have in my head is of a cylinder whose bottom edge has been stretched and twisted in order to be attached properly. But it's difficult for me to see how this could be the Möbius strip.

Any ideas?

Fred Byrd
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    It sounds like you're maybe trying to oversimplify things in your mind. Sure the M"obius strip has only one edge -- for example, you can glue opposite edges of a square to turn it into a torus and that has no edges. So it's easy to remove edges via edge-gluing constructions, especially when they're two-to-one. – Ryan Budney Aug 23 '13 at 17:12
  • The mapping cylinder in this case does not have two boundaries, since the top edge gets mapped to twice, you can think of it locally as having two sides. – Cheerful Parsnip Aug 23 '13 at 17:13
  • Indeed. I had a bad perspective on the image. – Fred Byrd Aug 23 '13 at 17:14

2 Answers2

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Neal
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There is always a down-to-earth approach. First note your mapping cylinder is glued from the usual cylinder $S^1\times [0,1]$, which could be in turn glued from a rectangle. Here is a picture how to do it, using a common way to describe glueing (the "notation" is explained in the end of the answer, but you probably can guess it):

rectangle

It's still far from obvious what the result would be. But now there is a standard trick for simplifying the task: do the auxilliary cuts. The point is, you will need to glue more (undoing the cut), but you can glue in different order. Let's cut the rectangle in half:

enter image description here results in enter image description here

Now glue horizontal (green) sides first:

enter image description here

This is nothing else but the Möbius band itself, done the standard way!

UPD: the meaning of pictures is as follows: take a rectangle, for each color (except usual black which doesn't mean anything) glue two straight segments with this color together the way directions of arrows coincide. In the first picture the bottom side is considered to be two segments.

Community
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Dmitry
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    I like your pictures. They show how to obtain the Mobius band from $S^1 \times [0,1]$. But what is the connection of your pictures to the given mapping $z \mapsto z^2$? It would be wonderful if you could add some explanation. – Manos Aug 26 '16 at 17:29
  • @Manos The gluing of the two green edges together corresponds to $z\mapsto z^2$. For that's just identifying $z\sim(-z)$ on the unit circle, and unravelling how the green edge gets turned into the base circle that appoints to $x\sim(x+1/2)$ which means you must stick the green edges together (giving Dmitry's final image) – FShrike Jan 11 '23 at 12:53