Riemann–Hurwitz formula for compact Riemann surfaces

(Properness and finiteness of the ramification.) By [F1], $f$ is proper, so [F2] applies. The branch values of $f$ form a finite set $B\subseteq Y$ because $Y$ is compact [F2]. The set $f^{-1}(B)$ is a finite union of finite fibres, hence finite. Every ramification point lies in $f^{-1}(B)$, although a fibre over a branch value can also contain unramified points. Thus ${\sum }_{x\in X}(e_x-1)$ has only finitely many nonzero terms.

(A topological triangulation with isolated branch vertices.) Apply [F3] with $F=B$ to obtain a finite face-to-face topological triangulation $\mathcal{T}$ whose edges avoid $B$. Insert each $b\in B$ into the open face containing it by joining it to that face's three vertices, choosing the arcs to avoid the other finitely many points of $B$ [F4]. Call the resulting triangulation ${\mathcal{T}}_0$; every branch value is now a vertex. Next choose one interior point on every edge of ${\mathcal{T}}_0$ and perform the edge subdivision of [F4], including the matching splits in its two incident faces. Call the final triangulation ${\mathcal{T}}^{\mathrm{\prime }}$ and its counts $V^{\mathrm{\prime }},E^{\mathrm{\prime }},F^{\mathrm{\prime }}$. Every old edge has been split, while an edge newly drawn during subdivision has its split point as an endpoint. Thus no edge of ${\mathcal{T}}^{\mathrm{\prime }}$ joins two vertices of ${\mathcal{T}}_0$, although a new edge may join two split points. In particular no edge, and hence no triangular face, has two branch-value vertices. Every subdivision preserves the topological cell count, so $V^{\mathrm{\prime }}-E^{\mathrm{\prime }}+F^{\mathrm{\prime }}=\chi (Y)$ by [F4] and [F6].

($f$ is a degree-$d$ covering off the vertex set.) Let $V^{\mathrm{\prime }}$ be the vertex set of ${\mathcal{T}}^{\mathrm{\prime }}$ and put $Y^{\ast }:=Y\setminus V^{\mathrm{\prime }}$. By step 2.1 the branch locus is contained in $V^{\mathrm{\prime }}$, so no point of $Y^{\ast }$ is a branch value; by [F2] the restriction $f^{\ast }:f^{-1}(Y^{\ast })\to Y^{\ast }$ is then a covering map of degree $d$: every point of $Y^{\ast }$ has an evenly covered neighbourhood on which $f$ is a $d$-sheeted covering.

(Vertices above the vertices.) For each $v\in V^{\mathrm{\prime }}$ the degree formula of [F2] gives ${\sum }_{x\in f^{-1}(v)}{e}_x=d$, and every $x$ there satisfies $e_x\ge 1$; hence $\mathrm{\mid }{f}^{-1}(v)\mathrm{\mid }=d-{\sum }_{x\in f^{-1}(v)}(e_x-1)$. Summing over the finitely many vertices $v\in V^{\mathrm{\prime }}$ gives $V_X=\mathrm{\mid }{f}^{-1}(V^{\mathrm{\prime }})\mathrm{\mid }=d{V}^{\mathrm{\prime }}-{\sum }_{x\in X}(e_x-1)$, because a point with $e_x>1$ is a ramification point and its image in $B$ is a vertex of ${\mathcal{T}}^{\mathrm{\prime }}$ by step 2.1, while a point with $e_x=1$ contributes nothing.

(Each open cell has exactly $d$ lifts.) Let $C\subseteq Y^{\ast }$ be the relative interior of an edge or an open face of ${\mathcal{T}}^{\mathrm{\prime }}$. An open edge is homeomorphic to the convex open interval, hence simply connected, and an open face is homeomorphic to an open triangle, which is convex, hence simply connected [F5]; a homeomorphism transports null-homotopies of loops, so both are simply connected in the sense of [F5]. Each connected component of $(f^{\ast }{)}^{-1}(C)$ is a connected covering of $C$, hence is one-sheeted over $C$ by [F5]; since every fibre of $f^{\ast }$ over $C$ has exactly $d$ points, $(f^{\ast }{)}^{-1}(C)$ has exactly $d$ components, each mapped homeomorphically onto $C$ by $f$.

(The lifted cells form a finite topological triangulation of $X$.) Fix a closed target triangle $T$ of ${\mathcal{T}}^{\mathrm{\prime }}$. By step 2.1, $T$ contains at most one branch value $v$, and if present it is a vertex. Put $T^{†}=T\setminus \{v\}$ in that case and $T^{†}=T$ otherwise. The homeomorphism from a Euclidean closed triangle in [F3] transports its straight-line contraction away from one vertex, so $T^{†}$ is connected and simply connected. The covering of $Y\setminus B$ supplied by [F2], restricted to $T^{†}$, therefore has $d$ components, each mapped homeomorphically to $T^{†}$ by [F5]. At an unramified vertex the local inverse of $f$ extends the sheet uniquely. At a branch vertex $v$, choose a small coordinate disk $D$ about $v$ such that $f^{-1}(D)$ is the disjoint union of coordinate disks about the finitely many points of $f^{-1}(v)$, with $f$ on each given by $z\mapsto z^{e_x}$; compactness of $X$ excludes further preimage components after shrinking $D$. The closed-triangle homeomorphism gives a neighbourhood basis of $v$ inside $T$ whose punctured members are connected. Take one such punctured corner neighbourhood contained in $D$. Its image under any one sheet inverse $T^{†}\to X$ is connected and therefore lies in one of those disjoint source disks, centred at some $x\in f^{-1}(v)$. As points of that corner approach $v$, the local equation $z^{e_x}=w$ forces their inverse images to approach $x$. Thus this sheet inverse extends continuously to $v$ by assigning it $x$. The extended map from compact $T$ to Hausdorff $X$ is injective, since it was injective on $T^{†}$ and $x$ lies over the omitted target point, and is therefore a homeomorphism onto a closed topological triangle. The same corner argument closes each lifted open edge to an interval, including at a ramified endpoint.

These closed lifted triangles meet face to face. Two different lifts of one target triangle have disjoint interiors and open edge lifts; by the local inverse at every unramified point they can meet only over its possible single branch vertex, hence in at most one vertex. For different target triangles, their images meet in at most one full target edge or one target vertex by [F3]. Over the interior of a common edge, the covering has exactly one lifted face on each local side of each lifted edge. An edge has at most one branch-value endpoint by step 2.1. If two closed lifted faces over adjacent target faces meet above both endpoints of that edge, they meet above an unramified endpoint; its unique local inverse makes their lifted edge germs coincide, and uniqueness of lifting along the connected open edge makes the whole lifted edge coincide. Thus they cannot meet in two vertices without sharing the full edge. Since each closed lifted triangle maps homeomorphically to its target triangle, two lifted triangles either share that one full lifted edge, share one vertex above a target vertex, or are disjoint. The local sectors of $z\mapsto z^{e_x}$ occur cyclically around a ramified source vertex, and ordinary local inverses give the usual circle links elsewhere. The lifted data therefore give a finite topological triangulation and hence a finite CW structure on $X$ by [F6]. There are $V_X:=\mathrm{\mid }{f}^{-1}(V^{\mathrm{\prime }})\mathrm{\mid }$ vertices, $E_X=d{E}^{\mathrm{\prime }}$ open edges and $F_X=d{F}^{\mathrm{\prime }}$ open faces by step 4.1, so $\chi (X)=V_X-E_X+F_X$.

(The cell count.) Combining steps 6.1 and 3.2, $\chi (X)=V_X-E_X+F_X=d{V}^{\mathrm{\prime }}-{\sum }_{x\in X}(e_x-1)-d{E}^{\mathrm{\prime }}+d{F}^{\mathrm{\prime }}=d(V^{\mathrm{\prime }}-E^{\mathrm{\prime }}+F^{\mathrm{\prime }})-{\sum }_{x\in X}(e_x-1)=d\chi (Y)-{\sum }_{x\in X}(e_x-1),$ using $\chi (Y)=V^{\mathrm{\prime }}-E^{\mathrm{\prime }}+F^{\mathrm{\prime }}$ from step 2.1.

(Substituting $\chi =2-2g$.) By [F6], $\chi (X)=2-2g(X)$ and $\chi (Y)=2-2g(Y)$; substituting into step 7.1 gives $2-2g(X)=d(2-2g(Y))-{\sum }_{x\in X}(e_x-1)$, and rearranging yields $2g(X)-2=d(2g(Y)-2)+{\sum }_{x\in X}(e_x-1)$.

(Conclusion.) The formula holds, and the ramification sum is finite by step 1.1. The Axiom of Choice is used exactly through the genus interface [F6] of [F7], that is, through the topological classification theorem on which it rests; the triangulation, its subdivision, the covering triviality over the simply connected cells and the local model are choice-free, every selection being made from finitely many explicit cells and fibres.