In Ferry and Ranicki's survey of the Wall finiteness obstruction (arxiv.org/abs/math/0008070) the following claim is made near the bottom of page 4:
For any maps $d:K\to X$, $s:X\to K$ there is defined a homotopy equivalence
$$T(d\circ s:X\to X)\to T(s\circ d:K\to K);\quad (x,t)\mapsto(s(x),t).$$
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:
$$T(s\circ d)\to T(d\circ s),\quad (k,t)\mapsto (d(k),t).$$
If the above is a genuine homotopy inverse, then the map:
$$(x,t)\mapsto(d(s(x)),t)$$
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 $T(s\circ d)\to T(d\circ s)$ 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?
Best Answer
Your guess for a homotopy inverse is correct. A homotopy between the identity and $(x,t) \mapsto (d(s(x)),t)$ is given by the expression $$ ((x,t), u) \mapsto \begin{cases} (x,t+u) & \text{ if } t+u\leq 1 \\ (d(s(x)), t+u-1) & \text{ if } t+u > 1 \end{cases} $$ where $u \in [0,1]$ is the parameter of the homotopy. Geometrically this shifts a point in the mapping torus of $d\circ s$ towards the right a distance of $u$ units.