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.
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
We have: (There might be an ambiguity and so
need not be well-defined.)
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
Then, by the homotopy lifting property, we can lift the homotopy
, establishing the claim.
It is clear from the construction that the map is a homomorphism: if
γ ( 1 ) = β ( 0 )
is the constant path at b.
τ ( [ β ] )
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.
One consequence of the construction is the below: