Galois Theory, Coverings, and Riemann Surfaces by Askold Khovanskii download in ePub, pdf, iPad
The action of Aut p on each fiber is free. This defines a group action of the deck transformation group on each fiber.
Note that by the unique lifting property, if f is not the identity and C is path connected, then f has no fixed points. If this action is transitive on some fiber, then it is transitive on all fibers, and we call the cover regular or normal or Galois.
The deck transformations are multiplications with n-th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group Cn. If x is in X and c belongs to the fiber over x i.
However the group G does act on the fundamental groupoid of X, and so the study is best handled by considering groups acting on groupoids, and the corresponding orbit groupoids. In this case the universal cover is also called the universal covering group. The universal cover first arose in the theory of analytic functions as the natural domain of an analytic continuation.
Deck transformations are also called covering transformations. This is known as the monodromy action.
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