# "Homotopy equivalence between two mapping tori of compositions For any maps d:K->X, s:X->K there is defined a homotopy equivalence T(d circ s:X->X)->T(s circ d:K->K);(x,t)->(s(x),t). Here, T(f) denotes the mapping torus of a self-map f:Z->Z (not necessarily a homeomorphism). It is very surprising to me that this holds with no extra conditions on d and s. I'm guessing that the homotopy inverse is the map: T(s circ d)->T(d circ s),(k,t)->(d(k),t). If the above is a genuine homotopy inverse, then the map: (x,t)->(d(s(x)),t) would have to be homotopic to the identity somehow. However, after banging my head against the wall on this for a while I can't come up with a valid homotopy. So my questions are: Is the map T(s circ d)->T(d circ s) I've defined above actually a homotopy inverse? If so,

Jaiden Elliott 2022-11-20 Answered
Homotopy equivalence between two mapping tori of compositions
For any maps $s:X\to K$ there is defined a homotopy equivalence
$T\left(d\circ s:X\to X\right)\to T\left(s\circ d:K\to K\right);\phantom{\rule{1em}{0ex}}\left(x,t\right)↦\left(s\left(x\right),t\right).$
Here, T(f) denotes the mapping torus of a self-map $f:Z\to Z$ (not necessarily a homeomorphism). It is very surprising to me that this holds with no extra conditions on d and s. I'm guessing that the homotopy inverse is the map:
$T\left(s\circ d\right)\to T\left(d\circ s\right),\phantom{\rule{1em}{0ex}}\left(k,t\right)↦\left(d\left(k\right),t\right).$
If the above is a genuine homotopy inverse, then the map:
$\left(x,t\right)↦\left(d\left(s\left(x\right)\right),t\right)$
would have to be homotopic to the identity somehow. However, after banging my head against the wall on this for a while I can't come up with a valid homotopy. So my questions are:
Is the map $T\left(s\circ d\right)\to T\left(d\circ s\right)$ I've defined above actually a homotopy inverse? If so, what is the homotopy from the composition I wrote down above to the identity map?
Is there a better one that makes the homotopy obvious?
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Miah Carlson
Your guess for a homotopy inverse is correct. A homotopy between the identity and $\left(x,t\right)↦\left(d\left(s\left(x\right)\right),t\right)$ is given by the expression

where $u\in \left[0,1\right]$ is the parameter of the homotopy. Geometrically this shifts a point in the mapping torus of d∘s towards the right a distance of u units.