Question

A Serre fibration has the homotopy lifting property with respect to the maps $[0,1]^n \times \{0\} \to [0,1]^{n+1}$. A Dold fibration $E \to B$ has the weak covering homotopy property: lifts with respect to maps $Y\times \{0\} \to Y \times [0,1]$ such that the lift agrees with the map $Y \to E$ up to a vertical homotopy (see the nLab page for more details. All Hurewicz fibrations are Dold fibrations, but not conversely, and not all Dold fibrations are Serre fibrations. I'm sure I read that not all Serre fibrations are Dold fibrations, but I don't have a counterexample.

My request is thus: an example of a Serre fibration that is not a Dold fibration.

Edit: I have found that a slight variant on this question was asked by Ronnie Brown in Proc. Camb. Phil.Soc. in October 1966, under the caveat that the base is path-connected and the base and the fibre have the homotopy type of a CW complex.

Answer 1

(answering my own question - who would have thought?)

There is a paper by G Allaud (Arch. Math 1968) which describes a counterexample as sought by the question. Let $E$ be the subspace of the plane $\mathbb{R}^2$ consisting of the non-negative integer points $(n,0)$ on the $x$-axis together with $(0,1)$ and a line connecting it each point on the $x$-axis. Let $B$ be the subspace of the plane consisting of the origin and the points $(1/n,0)$ on the $x$-axis for positive $n$ together with $(0,1)$ and a line connecting it to each point $(1/n,0)$ and $(0,0)$. The map $E \to B$ is given by sending $(0,0)$ to itself, $(n,0)$ to $(1/n,0)$ and the obvious map on the line segments. This is then (according to Allaud) a Serre fibration which is not a Dold fibration.