Change of fiber

In algebraic topology, 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" that starts at, say, "b", then we have the homotopy formula_1 where the first map is a projection. Since "p" is a fibration, by the homotopy lifting property, "h" lifts to a homotopy formula_2 with formula_3. We have:
Let formula_6 denote the set of path classes in "B". We claim that the construction determines the map:
Suppose β, β' are in the same path class; thus, there is a homotopy "h" from β to β'. Let
Drawing a picture, there is a homeomorphism formula_9 that restricts to a homeomorphism formula_10. Let formula_11 be such that formula_12, formula_13 and formula_14.
Then, by the homotopy lifting property, we can lift the homotopy formula_15 to "w" such that "w" restricts to formula_16. In particular, we have formula_17, establishing the claim.
It is clear from the construction that the map is a homomorphism: if formula_18,
where formula_20 is the constant path at "b". It follows that formula_21 has inverse. Hence, we can actually say:
Also, we have: for each "b" in "B",
which is a group homomorphism (the right-hand side is clearly a group.) In other words, the fundamental group of "B" at "b" acts on the fiber over "b", up to homotopy. This fact is a useful substitute for the absence of the structure group.
Consequence.
One consequence of the construction is the below: