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m →Riemann-Hurwitz formula: Fixing links to disambiguation pages, replaced: surface → surface using AWB |
Christian75 (talk | contribs) Clean up, typo(s) fixed: Therefore → Therefore, using AWB |
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That is because each simplex of ''S'' should be covered by exactly ''N'' in ''S''′ — at least if we use a fine enough [[Triangulation (geometry)|triangulation]] of ''S'', as we are entitled to do since the Euler characteristic is a [[topological invariant]]. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (''sheets coming together'').
Now assume that ''S'' and ''S′'' are [[Riemann surface]]s, and that the map π is [[analytic function|complex analytic]]. The map π is said to be ''ramified'' at a point ''P'' in ''S''′ if there exist analytic coordinates near ''P'' and π(''P'') such that π takes the form π(''z'') = ''z''<sup>''n''</sup>, and ''n'' > 1. An equivalent way of thinking about this is that there exists a small neighborhood ''U'' of ''P'' such that π(''P'') has exactly one preimage in ''U'', but the image of any other point in ''U'' has exactly ''n'' preimages in ''U''. The number ''n'' is called the ''[[ramification index]] at P'' and also denoted by ''e''<sub>''P''</sub>. In calculating the Euler characteristic of ''S''′ we notice the loss of ''e<sub>P</sub>'' − 1 copies of ''P'' above π(''P'') (that is, in the inverse image of π(''P'')). Now let us choose triangulations of ''S'' and ''S′'' with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then ''S′'' will have the same number of ''d''-dimensional faces for ''d'' different from zero, but fewer than expected vertices. Therefore, we find a "corrected" formula
:<math>\chi(S') = N\cdot\chi(S) - \sum_{P\in S'} (e_P -1) </math>
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