If $E$ maps onto a contractible space with contractible fibers, must $E$ be contractible? 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.
 A: 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
  
  
*
  
*$Y$ is $LC^n$ and 
  
*$f$ is an $n$-equivalence.
  

Some comments:


*

*$f$ is proper means the preimages of compact sets are compact; this is automatic if $X$ is compact and $Y$ is Hausdorff.

*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.
A: 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 
