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

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.

-
Do you want $F$ to be $\pi^{-1}(x)$? Or is $F$ some fixed space independent of $x$? –  John Pardon Dec 18 '11 at 19:49

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 $U$.

-