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This is probably obvious to topologists so I'll just come right out with the question:

What is an example of a surjective homotopy equivalence $E \to B$ of path-connected CW complexes which is not a Serre fibration?

The context in which I'm asking is that I wanted to know if there were any equivalent definitions of Serre fibration. The first thing I looked for was if having homotopy equivalent fibers was sufficient - I didn't find a counterexample, but I gather it's not.

I read that all Serre fibrations are "homotopy fiber bundles" (cf. What is the relation between a ''homotopy fiber bundle'' and a Serre fibration?). However, the answer there mentions that the converse is not true:

There are many "homotopy fiber bundles" which are not fibrations. For example take any homotopy equivalence $E\to B$ that is not a fibration, then it is a "homotopy fiber bundle" with a one-point fiber.

The first example of an h.e. which is not a fibration that came to mind was a deformation retract $A \hookrightarrow X$. Since fibrations over a path-connected base have h.e. fibers, they're necessarily surjective (or empty/initial), which makes this counterexample a little artificial. What I mean by that is that it's easy to modify the definition of homotopy fiber bundle adding a surjectivity assumption to exclude examples like this. For instance, I don't know whether "surjective homotopy fiber bundle" is equivalent to "(non-empty) Hurewicz fibration" though I assume it's not.

Essentially, the purpose of the question is to ask if someone can help me better understand the difference between homotopy fiber bundles and fibrations. Answers to variations of the question are also welcome (e.g. Hurewicz fibration version).

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  • $\begingroup$ Take the + sign and project to the - sign. The homotopy lifting property fails already for a point (aka, the path-lifting property). $\endgroup$ – user98602 Jan 16 at 8:25
  • $\begingroup$ @MikeMiller thank you! $\endgroup$ – Ben Jan 16 at 11:13
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Any map $E \to B$ such that there is a fiber $F_b$ containing a point $x \in F_b$ which is isolated from other nearby fibers will fail the path lifting property.

Thank you Mike Miller for providing the example in the comment.

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