[Math] Mapping torus of a homotopy equivalence

Question

The mapping torus $M_f$ of a homeomorphism $f$ of some topological space $X$ is a fiber bundle whose base is a circle and whose fiber is the original space $X$. If instead of a homeomorphism $f$ is just a homotopy equivalence of $X$, is $M_f$ a fibration over the circle with fiber homotopic to $X$?

Answer 1

There's no reason to expect $M_f$ to be a fiber bundle, or even a Hurewicz or Serre fibration. (Think of $f: \mathbb{R} \to \mathbb{R}$ by $f(x)=0$. This is a homotopy equivalence, but $M_f$ is certainly not locally trivial, nor does $M_f \to S^1$ have any nice lifting properties.)

What is true is that the homotopy fiber of $M_f\to S^1$ is weakly equivalent to $X$, if $f$ is a homotopy equivalence (or even a weak equivalence). This often gets proved by using the theory of "quasi-fibrations".