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Let $p\colon E\to C$ be a continuous, surjective map between topological spaces with $C$ contractible. Suppose that $p^{-1}(c)$ is contractible for each $c\in C$. Is it true that $E$ is weakly contractible?

If not, what are some mild conditions that will guarantee that this is true? I am aware that this is true if $p$ is a (quasi-)fibration. Sorry if this is an easy question.

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For a less pathological counterexample, consider the obvious bijection $[0,1]\cup (2,3]\to [0,2]$. – Eric Wofsey Jun 20 '14 at 12:40
Look up the Vietoris–Begle mapping theorem. – Jeff Strom Jun 20 '14 at 13:30

2 Answers 2

Here is the main theorem of "A Vietoris Mapping Theorem for Homotopy" by S. Smale:

THEOREM: Let $f:X\to Y$ be proper and onto, where $Y$ and $X$ are

  • $0$-connected
  • separable
  • locally compact
  • metric.

Assume $X$ is $LC^n$ (which I presume means locally $n$-connected) and that for each $y\in Y$, $f^{-1}(Y)$ is $LC^{n-1}$ and $(n-1)$-connected. Then

  1. $Y$ is $LC^n$ and
  2. $f$ is an $n$-equivalence.

Some comments:

  1. $f$ is proper means the preimages of compact sets are compact; this is automatic if $X$ is compact and $Y$ is Hausdorff.
  2. the local conditions on $X$ and $Y$ are satisfied if $X$ and $Y$ are CW complexes.

In your situation, if your $f$ satisfies the conditions of this theorem for all $n$, then we deduce that $f$ is an $n$-equivalence for all $n$ and hence that $E$ is weakly contractible because $C$ is contractible.

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Check out:

MR0036509 (12,120e) Reviewed Borel, Armand; Serre, Jean-Pierre Impossibilité de fibrer un espace euclidien par des fibres compactes. (French) C. R. Acad. Sci. Paris 230, (1950). 2258–2260. 56.0X

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