# A morphism of fiber bundles induces a continuous map of the fibers

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?

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

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.
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 $$GL_n(\mathbb{R})$$.