Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\pi(x)$ 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.

share|improve this question
add comment

1 Answer

A Serre fibration is defined by a lifting property against maps $X \to X \times [0;1]$ for cubes $X$. There is some kind of homotopy version of this, which is called the "weak covering homotopy property" (WHCP). Maps with the WHCP for all spaces are called Dold fibrations and these turn out to be "locally homotopy trivial maps", such as your homotopy fiber bundle.

Check out http://ncatlab.org/nlab/show/Dold+fibration for more detailled definitions.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.