Change of fiber

Given a fibration p:E→B, the change of fiber is a map between the fibers induced by paths in B.

Since a covering is a fibration, the construction generalizes the corresponding facts in the theory of covering spaces.

Definition

If β is a path in B, then we have the homotopy $h:p^{-1}(b)\times I\to I{\overset {\beta }{\to }}B$ where the first map is a projection. Since p is a fibration, by the homotopy lifting property, h lifts to a homotopy $g:p^{-1}(b)\times I\to E$ with $g_{0}:p^{-1}(b)\hookrightarrow E$. We have:

$g_{t}:p^{-1}(b)\to p^{-1}(\beta (t))$.

We let $\tau (\beta )=g_{1}$. Let $\operatorname {Pc} (B)$ denotes the set of path classes in B. We claim that the τ we have just constructed induces:

$\tau :\operatorname {Pc} (B)\to$ the set of homotopy classes of maps.

Suppose β, γ are in the same path class; thus, there is a homotopy h from β to γ. Thinking I² is a cube, let K be its subset obtained by removing the edge corresponding to.

It is immediate from the construction that the map is a homomorphism:

$\tau ([\beta ]\cdot [\gamma ])=\tau ([\beta ])\tau ([\gamma ])$.

Consequence

One can get a substitute for a structure group. Indeed, suppose B is path-connected. Then