In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.
Statement For a compact, connected, orientable surface S {\displaystyle S} , the Euler characteristic χ ( S ) {\displaystyle \chi (S)} is
χ ( S ) = 2 − 2 g {\displaystyle \chi (S)=2-2g} , where g is the genus (the number of handles). This follows, as the Betti numbers are 1 , 2 g , 1 , 0 , 0 , … {\displaystyle 1,2g,1,0,0,\dots } . For the case of an (unramified) covering map of surfaces
π : S ′ → S {\displaystyle \pi \colon S'\to S}
that is surjective and of degree N {\displaystyle N} , we have the formula
χ ( S ′ ) = N ⋅ χ ( S ) . {\displaystyle \chi (S')=N\cdot \chi (S).}
That is because each simplex of S {\displaystyle S} should be covered by exactly N {\displaystyle N} in S ′ {\displaystyle S'} , at least if we use a fine enough triangulation of S {\displaystyle 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 {\displaystyle S} and S ′ {\displaystyle S'} are Riemann surfaces, and that the map π {\displaystyle \pi } is complex analytic. The map π {\displaystyle \pi } is said to be ramified at a point P ′ {\displaystyle P'} in S ′ {\displaystyle S'} if there are analytic coordinates in open neighboorhoods U ′ {\displaystyle U'} of P ′ {\displaystyle P'} and U = π ( U ′ ) {\displaystyle U=\pi (U')} near P = π ( P ′ ) {\displaystyle P=\pi (P')} such that π {\displaystyle \pi } takes the form π ( z ) = z n {\displaystyle \pi (z)=z^{n}} with n > 1 {\displaystyle n>1} . Equivalently, the point P {\displaystyle P} has exactly one nearby preimage π − 1 ( P ) ∩ U ′ = { P ′ } {\displaystyle \pi ^{-1}(P)\cap U'=\{P'\}} , but any other point Q {\displaystyle Q} in U has exactly n nearby preimages π − 1 ( Q ) ∩ U ′ = { Q 1 ′ , … , Q n ′ } {\displaystyle \pi ^{-1}(Q)\cap U'=\{Q'_{1},\ldots ,Q'_{n}\}} . The number n is called the ramification index at P ′ {\displaystyle P'} and is denoted by e P ′ {\displaystyle e_{P'}} . In calculating the Euler characteristic of S ′ {\displaystyle S'} we notice the loss of e P ′ − 1 {\displaystyle e_{P'}-1} copies of P ′ {\displaystyle P'} above P {\displaystyle P} . Now compute the Euler characteristics using triangulations of S {\displaystyle S} and S ′ {\displaystyle S'} with vertices at the respective branch and ramification points. The triangulation of S ′ {\displaystyle S'} will have the same number of positive-dimensional faces as in the unramified case, but fewer than expected vertices. The correct formula accounting for ramifications is the Riemann–Hurwitz formula or Hurwitz's theorem:
… excerpt ends here. Continue reading the full article.
