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Given a fiber bundle $p:E\to B$ and a point $x\in B$, is the evaluation map $\varepsilon:\Gamma^0(E)\to p^{-1}(x)$ a weak homotopy equivalence when $\Gamma^0(E)$ is endowed with the compact-open topology? I'm having trouble coming up with a good counter example but also failed at assuming it was a weak homotopy equivalence in general and trying to prove it (which could simply hint at me being bad at this).

UPDATE: Right now it seems unlikely to me that the evaluation maps are w.h.e.'s in general since surjectivity of the induced maps on homotopy doesn't seem to work in general since there are no global sections of $E$ in general, or am I completely off track right now?

EDIT: Just to make it clear, $\varepsilon:\Gamma^0(E)\to p^{-1}(x)$ is defined by $\varepsilon(\sigma):=\sigma(x)$.

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It's not a good idea to post both here and on MO, unless you've waited somewhat longer in between posts. In any event, look at my answer on MO: mathoverflow.net/questions/69024/… –  Dan Ramras Jun 28 '11 at 16:46

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As Dan Ramras pointed out on MO the statement in the first paragraph is quite obviously (although not to me in the past) false in general. Instead if one additionally assumes contractibility of the base space one obtains very easily the result that evaluation maps for sections are indeed (weak) homotopy equivalences.

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