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Let $E$, $B$, $F$ be topological spaces, and $p:E\to B$ a continuous surjection. According to my course, this data define a fiber boundle if, for any $b\in B$, exist a open neighborhood $U$ of $b$ and a homeomorphism $h:p^{-1}(U)\to U\times F$ such that (letting $\pi:U\times F\to U$ be the canonical projection and writing $p_\cdot$ for the map $p^{-1}(U)\to U$ induced by $p$) $p_\cdot=\pi\circ h$.

Observation for later: in the setting above, $h$ restricts to a homeomorphism from $p^{-1}(b)$ to $\{b\}\times F\cong F$.

If $p:E\to B$ and $p':E'\to B'$ are fiber bundles, with fibers $F$ and $F'$, a morphism consists of continuous maps $\phi:B\to B'$ and $\phi':E\to E'$, such that $p'\circ \phi'=\phi\circ p$. Here the notes point out that these data also give a continuous map $F\to F'$.

One way I'd recover a map $F\to F'$ is as follows. Fix $x\in B'$ and $y\in \phi^{-1}(x)$. Then $p^{-1}(y)\cong F$ and $p'^{-1}(x)\cong F'$; also, $\phi'$ restricts to $p^{-1}(y)\to p'^{-1}(x)$, giving a continuous map $F\to F'$ via the homeomorphisms.

My question is: since this map $F\to F'$ doesn't seem unique, but rather depends on $x$ and $y$, are there any conditions of commutativity / coherence, letting $x,y$ vary, that should be known?

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  • $\begingroup$ I believe in the second last paragraph $y$ should be from $\phi^{-1}(x)$. $\endgroup$ Dec 13, 2022 at 16:12

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Firstly, let me propose a piece of notation that might make things more clear. Let us denote by $F_x$ the fiber in $E$ on $x\in B$, and similarly for $F'_y$ in $E'$. If $\phi(x)=y$, then the restriction of $\phi'$ to $F_x$ is the map from $F_x$ to $F'_y$ that you are looking for.

As for the conditions on these maps, I should say maps between spaces with different fibers and base spaces don't come up that much. You will see that in many books they define a bundle map as a map (as you defined it) together with the condition that the base spaces are the same. Otherwise, the map in question will most likely be a pullback bundle, in which case the fibers are the same.

If you want to consider all the maps from $F$ to $F'$ that can be constructed using a given $\phi$ and $\phi'$, then you will need to use the homeomorphisms between $p^{-1}(U)$ and $U\times F$ since different homeomorphisms (say from $U$ and $V$ in the base space with nonempty intersection) will give rise to different maps between fibers. If you are interested in these and the conditions we put on local trivializations of $p^{-1}(U)$, then you should look into vector bundles and principal bundles. For example, in the case of vector bundles, the fibers are homeomorphic to $\mathbb{R}^n$. Given two trivializing neighbourhoods $U$ and $V$ in $B$ with a common point $x$, we require that the map from $F_x$ to itself induced by the trivializations $p^{-1}(U)\cong U\times F$ and $p^{-1}(V)\cong V\times F$ is linear, i.e. in $GL_n(\mathbb{R})$.

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