## What is the relation between a ‘'homotopy fiber bundle’' and a Serre fibration?

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Since I got no responses to this question at Stack Exchange, please let me try my luck here.

Call a continuous map $\pi:E\to B$ between CW complexes a homotopy fiber bundle if for any $x$ in the image of $\pi$, there is an open neighbourhood $U\subset B$ of $x$ and homotopy equivalence $\pi^{-1}(U)→U\times F$ over $U$.

I don't know if this has a different name in the literature or even if it is reasonable. Replacing ''homotopy equivalence'' by ''homeomorphism'' should be the definition of an ordinary fiber bundle.

How relates a ''homotopy fiber bundle'' to the notion of a Serre fibration?

At least both properties imply that the fibers over connected components are all weakly homotopy equivalent.

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Do you want $F$ to be $\pi^{-1}(x)$?  Or is $F$ some fixed space independent of $x$? – unknown (google) Dec 18 2011 at 19:49

## 2 Answers

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.

On the other hand the other implication is (almost) true. The following works for Hurewicz fibrations. I don't whether it is true that a fiber of a Serre fibration between CW-complexes has a homotopy type of a CW-complex. If this is the case, then the proof works also for Serre fibrations.

Let $\pi : E \to B$ be a Hurewicz fibration between CW-complexes and $x \in B$. CW-complexes are locally contractible, so there is a contractible neighborhood $U$ of $x$. Let $\pi_U : E_U \to U$ be the restriction of $\pi$ to $U$. If $E_x$ is the fiber of $\pi$ at $x$, then the inclusion $E_x \to E_U$ is a pullback of the inclusion `$\{x\} \to U$` along a Hurewicz fibration, so it is a homotopy equivalence and admits a homotopy inverse $f : E_U \to E_x$. Thus $(\pi_U, f) : E_U \to U \times E_x$ is a homotopy equivalence over $U$.

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There is a nice theorem in geometric topology that a Serre fibration between actual CW complexes (not just homotopy types thereof) is in fact a Hurewicz fibration. It's in a paper by Steinberger and West (Covering homotopy properties of maps between CW complexes or ANRs, Proc AMS92(1984), 573-577), with a correction in a paper by R. Cauty (Sur les ouverts des CW-complexes et les fibr\'es de Serre, Colloq. Math. 63(1992), 1-7).

Then the rest of your answer applies. But of course the proof of the cited theorem proceeds by checking homotopical local triviality and then quoting the local to global characterization of Hurewicz fibrations.

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