Change of fiber

Given a fibration p:E→B, the change of fiber is a map between the sets of path classes 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, since p is a fibration, there exists a homotopy

Let $Pa(B)$ denotes the set of path classes in B. Then we thus get a map

$\tau :Pa(B)\to$

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.