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Topological Galois Theory pp 107–141Cite as

Coverings and Galois Theory

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Abstract

This chapter is devoted to the geometry of coverings and its relation to Galois theory. There is a surprising analogy between the classification of coverings over a connected, locally connected, and locally simply connected topological space and the fundamental theorem of Galois theory. We state the classification results for coverings so that this analogy becomes evident.

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Notes

  1. 1.

    A point y that is an analytic-type singularity is also called a topological branch point.

  2. 2.

    An analytic-type map is also called a topological branched covering.

  3. 3.

    A singular point is also called a critical point.

  4. 4.

    Galois theory allows one to obtain the following result. Suppose that the answer for a subgroup H is positive, and let f a  ∈ P a (O) be a germ whose stabilizer is equal to H. Let \(\tilde{H}\) denote the largest normal subgroup lying in H. Then for every subgroup containing the group \(\tilde{H}\), the answer is also positive. For the proof, it suffices to apply the fundamental theorem of Galois theory to the minimal Galois extension of the field K(X) containing the germ f 1.

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  1. Department of Mathematics, University of Toronto, Toronto, Ontario, Canada

    Askold Khovanskii

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Khovanskii, A. (2014). Coverings and Galois Theory. In: Topological Galois Theory. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38871-2_4

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