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Let $B$ be a finite CW complex and $p:E\to B$ a Hurewicz fibration.

I know that every topological space $E$ has the weak(!) homotopy type of a CW complex $D$ realized by a weak equivalence $f:D\to E$. (I apologize for asking two things in one question but they are so related that splitting them up seems a little exaggerated.)

  1. Is it possible to find a CW complex $D'$, a Hurewicz fibration $p':D'\to B$ and a weak equivalence $f':D'\to E$ such that $p'=p\circ f'$?

  2. Is it possible to find a Serre fibration $q:C\to B$ and a weak equivalence $g:C\to E$ such that $q=p\circ g$? Can I choose $C$ to be a CW complex?

ADDENDUM:

Due to Henry T. Horton's great answer, the whole question reduces to the following question.

Let $E$ be a topological space. Is it possible to find a CW complex $D$ and a weak equivalence $f:D\to E$ such that $f$ is also a Serre fibration?

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1 Answer 1

For the second question, consider the following. Let $$B^I \longrightarrow B,$$ $$\alpha \mapsto \alpha(0)$$ be the path space fibration and consider the pullback \begin{matrix} p^\ast B^I & \longrightarrow & E \\ \downarrow & & \downarrow \\ B^I & \longrightarrow & B \end{matrix} Write $P_p = p^\ast B^I$; we call $P_p$ the mapping path space of $p: E \longrightarrow B$. We have the mapping path fibration $$q: P_p \longrightarrow B,$$ $$(e, \alpha) \mapsto \alpha(1).$$

Theorem. $q: P_p \longrightarrow B$ is a Serre fibration and there exists a homotopy equivalence $g: P_p \longrightarrow E$ such that $q = p \circ g$. In fact, since $p$ is a fibration, $g$ is a fiber homotopy equivalence.

The map $g: P_p \longrightarrow E$ above is actually just the projection onto the first factor: $$P_p \longrightarrow E,$$ $$(e, \alpha) \mapsto e.$$ See the proof of Theorem 6.18 of Davis and Kirk's Lecture Notes in Algebraic Topology (available here) for the details on why $q$ is a Serre fibration.

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Thanks for pointing this out. The whole question would be answered positively, if a CW approximation $D\to E$ of an arbitrary topological space $E$ could be chosen as a Serre fibration. Is this possible? –  Magneto Jan 23 '13 at 8:08

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