nLab
branched cover

Context

Geometry

Bundles

Contents

Idea

A “branched” or “ramified” covering spaces is much like a plain covering space, only that over isolated points in the base space – called the branch point or ramification points – sheets of the covering may merge.

The archetypical examples in complex analysis and here the most archetypical example of all is the graph of (any choice of) the square root function on the complex plane, which is a double cover away from the point at the origin, but including the origin it is a branched double cover with branch point that origin

Dually branching is reflected in rings of functions by ramification of ideals.

Notably under the function field analogy one may also understand ramification of ideals in number fields as encoding branched coverings (“over Spec(F1)”)

Examples

References

  • Wikipedia, Branched covering

  • Jürgen Neukirch, Algebraische Zahlentheorie (1992), English translation Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften 322, 1999 (pdf)

Last revised on July 23, 2014 at 06:43:48. See the history of this page for a list of all contributions to it.