This may be a weird question but I really want to know the answer:
Hatcher p.2:
Why is the name 'cylinder' used in this instance? I don't think that this quotient space, namely the mapping cylinder, is homeomorphic to a cylinder. At the beginning of the chapter, he says that "... it should be read in this informal spirit, skipping bits here and there." So should I just ignore this?
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$\begingroup$First, one constructs a cylinder over the space $X$, that is, $X\times I$ (here $I=[0,1]$). Then one maps the top of the cylinder into $Y$. You can think on it as gluing the top $X\times \{1\}$ with the image $f(X)$.
$\endgroup$ 5 $\begingroup$It is called a cylinder only really to help with intuition. It is only homeomorphic to an actual cylinder ($S^1\times I$) when we have a map $f\colon S^1\rightarrow S^1$ from the circle to itself, and $f$ is a homeomorphism.
You will later also come across the notion of a mapping torus which is a mapping cylinder of a map from a space to itself, but where we then identify the 'boundary space' with respect to the map. That is, if $f\colon X\rightarrow X$ is a continuous map, then the mapping torus $\mathcal{M}_f$ is the space $(X\times I)/(x,0)\sim(f(x),1)$.
In the same way, this construction is only homeomorphic to an actual torus if we have a map $f\colon S^1\rightarrow S^1$ from the circle to itself, and $f$ is a homeomorphism.
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