Change of fiber

In algebraic topology, given a fibration p:EB, 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:

.

(There might be an ambiguity and so need not be well-defined.)

Let denote the set of path classes in B. We claim that the construction determines the map:

the set of homotopy classes of maps.

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:

the set of homotopy classes of homotopy equivalences.

Also, we have: for each b in B,

{ [ƒ] | homotopy equivalence }

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:

  • The fibers of p over a path-component is homotopy equivalent to each other.

References

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