Question

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.

Answer 1

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.

Answer 2

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.