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 $X$?
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 "quasifibrations". 


Check out Homotopy equivalences and mapping torus projections D. S. Coram, P. F. Duvall Fund. Math. 109 (1980), 17 


This is closely related to a question of mine, which was motivated by wondering whether the mapping cylinder of a homotopy equivalence is a fibration over an interval. The counterexample given there (for the homotopy equivalence from I to a point) also gives a counterexample for the mapping torus, and makes it easy to see how it goes wrong. 

