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 where the first map is a projection. Since p is a fibration, by the homotopy lifting property, h lifts to a homotopy with. We have:
Let 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 that restricts to a homeomorphism. Let be such that, and.
Then, by the homotopy lifting property, we can lift the homotopy to w such that w restricts to. In particular, we have, establishing the claim.
It is clear from the construction that the map is a homomorphism: if,
where is the constant path at b. It follows that has inverse. Hence, we can actually say:
Also, we have: for each b in B,
which is a group homomorphism 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: