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Suppose that for a contractible space $A$ we are given a fiber bundle $p:E\to A$ and denote for $B\subset A$ by $E(B)$ the restricted bundle. I have good reason to believe that in this situation the restriction map $r:\Gamma^0(E)\to\Gamma^0(E(B))$ is a Serre fibration when the section spaces are endowed with the compact-open topology. After a few unsuccessful tries of proving this I decided it might not be a bad idea to ask the denizens of StackExchange for some help. Any hints/ideas/comments are very appreciated.

EDIT: To give a concrete example that bothers me consider $A=D^k\times D^{m-k}$ and $B=D^k_{\frac{1}{2}}\times D^{m-k}$ where $D^k$ denotes the unit disc in $k$ dimensions and $D^k_{\alpha}:=\{x\in D^k\,\big|\,\alpha\leq\|x\|\leq 1\}$.

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    Here are some hints. In your example, $A$ is contractible, so you can reformulate the problem more concretely. Also, the inclusion of B into A is a cofibration. (Once you figure out the answer to your question, you'll understand the term "cofibration.") For general (non-contractible) base spaces, I'd worry about sections over B that don't extend to sections over all of A.2011-07-03
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    @Dan Thank you for the hints, you're starting to become my guardian angel. What I got now is that assuming in my concrete example that the inclusion of $B$ into $A$ is indeed a cofibration it seems quite easy to show that $r$ is a Serre fibration using the facts that $A$ is contractible and the identification $\Gamma^0(E)=C^0(A,F)$ for a suitable fiber $F$. So I'd be correct in assuming that the statement of the first paragraph is correct if I additionally assume that the inclusion for general $B$ in $A$ is a cofibration?2011-07-03
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    Additionally I want to say that I haven't yet managed to construct a retraction of $A\times [0,1]$ onto $(A\times\{0\}\cup B\times [0,1])$ for the concrete $A$ and $B$ I gave. I'm sure it must be something simple but it just goes to show how bad I am at computation (amongst other things).2011-07-03
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    Chapter 0 of Hatcher's Algebraic Topology is a good place to read about cofibrations. In particular, any subcomplex of a CW complex gives a cofibration. Writing down the desired maps explicitly can certainly be a pain.2011-07-03
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    And yes, the first paragraph looks correct once you assume you have a cofibration.2011-07-03
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    Ah, good ol' Hatcher. I seem to underestimate that book heavily, it's always the last place I consult. Thank you for the help again, figured the appropriate retraction out in the mean time. I'll either write an answer later or just delete the question. Or you might just copy one of your comments into an answer if you're after any of these points they got here, I'll accept gladly.2011-07-03

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In your example, A is contractible, so you can reformulate the problem more concretely. Also, the inclusion of B into A is a cofibration, which implies that the dual map restriction map on mapping spaces is a fibration.