Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The following is a result I feel like I've seen some form of before, but can't figure out how to prove or find a reference for. Suppose you have a map p:E \to B, with B paracompact, and suppose that every point in B has a neighborhood U such that there is a map p^{-1}(U) \to U \times F over U which is a fiber homotopy equivalence. Does it follow that p is a Hurewicz fibration? The converse, if B is locally contractible, is standard: a Hurewicz fibration is locally equivalent to a product.

share|improve this question
add comment

2 Answers

up vote 3 down vote accepted

The answer is no; Allen Hatcher sent me the following:

An example where this fails is the projection of the letter L onto its horizontal base, which I'll call B. The deformation retraction of L onto B is a fiberwise homotopy equivalence. The homotopy lifting property fails: Map a point to the left endpoint of B, then lift this to a point of L - B and take a homotopy that moves the left endpoint of B to the right endpoint.

share|improve this answer
add comment

I think this can be extracted from Spanier's book. In Chapter 2.7, his Theorem 13 says that if B is a paracompact Hausdorff space, then a map p:E-->B is a fibration if and only if it is a local fibration. By a local fibration, he means that there is a covering {U_{\alpha}} of B such that for all \alpha, the map p restricted to p^{-1}(U_{\alpha}) is a fibration. I'm pretty sure that the condition you describe implies this.

share|improve this answer
Unfortunately, the local property fails to imply it's a fibration; see my answer. –  Eric Wofsey Oct 27 '09 at 17:52
add comment

Your Answer


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.