"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 K there is defined a homotopy equivalence
T ( d s : X X ) T ( s 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 d ) T ( d 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 d ) T ( d s ) 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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Answers (1)

Miah Carlson
Answered 2022-11-21 Author has 17 answers
Your guess for a homotopy inverse is correct. A homotopy between the identity and ( x , t ) ( d ( s ( x ) ) , t ) is given by the expression
( ( x , t ) , u ) { ( x , t + u )  if  t + u 1 ( d ( s ( x ) ) , t + u 1 )  if  t + u > 1
where u [ 0 , 1 ] 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.
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