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Change of fiber

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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

Consequence

One can get a substitute for a structure group.

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