# Is the restriction map of sections of a fiber bundle a Serre fibration?

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. – Dan Ramras Jul 3 '11 at 6:01
@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? – Martin Worsek Jul 3 '11 at 9:07
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). – Martin Worsek Jul 3 '11 at 9:09
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. – Dan Ramras Jul 3 '11 at 22:19
And yes, the first paragraph looks correct once you assume you have a cofibration. – Dan Ramras Jul 3 '11 at 22:20

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.

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