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?
Best Answer
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