Define the mapping torus of a homeomorphism [itex]\phi:X \rightarrow X[/itex] to be the identification space(adsbygoogle = window.adsbygoogle || []).push({});

[itex]T(\phi)= X \times I / \{ (x,0) \sim (\phi(x),1) | x \in X \}[/itex]

I have to identify [itex]T(\phi)[/itex] with a standard space and prove that it is homotopy equivalent to [itex]S^1[/itex] by constructing explicit maps [itex]f:S^1 \rightarrow T(\phi), g: T(\phi) \rightarrow S^1[/itex] and explicit homotopies [itex]gf \simeq 1:S^1 \rightarrow S^1, fg \simeq 1:T(\phi) \rightarrow T(\phi)[/itex] in the two cases:

(i) [itex]\phi(x)=x[/itex] for [itex]x \in X=I[/itex]

(ii) [itex]\phi(x)=1-x[/itex] for [itex]x \in X=I[/itex]

i found that since [itex]X=I[/itex], we have a square of side 1 to consider:

in (i) we identify two opposite sides with one another, this gives us a cylinder.

in (ii) we identify the point x with the point 1-x on the opposite side giving a kind of "twist" which i think leads to a Mobius strip.

first of all, are my answers above correct? it says to identify them with a standard space. is there some sort of notation i can use for cylinders and Mobius strips? e.g. i can call a circle [itex]S^1[/itex], is there something like [itex]C^1[/itex] for a cylinder?

then, how do i go about setting up the maps [itex]f[/itex] and [itex]g[/itex]?

thanks

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# Homework Help: Define the mapping torus of a homeomorphism

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