Vipul: 1 revision

2008-05-18T19:18:10Z

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Vipul: New page: ==Statement== Suppose $M,N$ are compact Riemann surfaces and $f:M \to N$ is an analytic map. Then we have: $g_M - 1 = \operatorname{deg}(f)(g_N - 1) + ...$

2008-05-03T22:26:18Z

New page: ==Statement== Suppose <math>M,N</math> are compact Riemann surfaces and <math>f:M \to N</math> is an analytic map. Then we have: <math>g_M - 1 = \operatorname{deg}(f)(g_N - 1) + ...

New page

==Statement==

Suppose <math>M,N</math> are [[compact Riemann surface]]s and <math>f:M \to N</math> is an [[analytic map]]. Then we have:

<math>g_M - 1 = \operatorname{deg}(f)(g_N - 1) + \frac{1}{2}B</math>

where <math>B</math>, termed the [[branching number]] of <math>f</math>, is defined as:

<math>B := \sum_{p \in M} (v_f(p) - 1)</math>

where <math>v_f(p)</math> is the local degree of the map <math>f</math> around <math>p</math>. Note that the set of <math>p</math> for which <math>v_f(p) \ne 1</math> is a finite set, because it is discrete and closed and <math>M</math> is a compact set. The points <math>p</math> for which <math>v_f(p) \ne 1</math> are termed ''branch points''.

Note that in case <math>B = 0</math>, we get an actual covering map, and in this case we are simply told that the degree of the covering map equals the ratio of the Euler characteristics of the Riemann surfaces.

Riemann-Hurwitz formula - Revision history

Vipul: 1 revision

Vipul: New page: ==Statement== Suppose M,N are compact Riemann surfaces and f:M \to N is an analytic map. Then we have: g_M - 1 = \operatorname{deg}(f)(g_N - 1) + ...

Vipul: New page: ==Statement== Suppose $M,N$ are compact Riemann surfaces and $f:M \to N$ is an analytic map. Then we have: $g_M - 1 = \operatorname{deg}(f)(g_N - 1) + ...$