Genus and Euler characteristic of a compact Riemann surface

Assume the Axiom of Choice. Let $X$ be a compact Riemann surface (Riemann surfaces and holomorphic atlases), and let $X\cong {\mathrm{\#}}_g{T}^2$ be its homeomorphism type as a sphere with $g$ handles (Topological classification of compact Riemann surfaces); here ${\mathrm{\#}}_g{T}^2$ is the connected sum of $g$ copies of the torus and ${\mathrm{\#}}_0{T}^2=S^2$.

• The genus of $X$ is that number, $g(X):=g\ge 0.$ Equivalently, $g(X)=0$ exactly when $X$ is homeomorphic to the sphere, and $g(X)\ge 1$ is the unique number of handles in the classification.
• The Euler characteristic of $X$ is the alternating cell count of any finite cell structure of $X$, $\chi (X):=V-E+F,$ in the sense of Euler characteristic of a finite CW complex: choose a finite triangulation or polygonal schema of $X$, count its vertices, edges and faces, and take the alternating sum.

The number $g(X)$ is well defined by the uniqueness clause of Topological classification of compact Riemann surfaces: among the orientable normal forms ${\mathrm{\#}}_g{T}^2$, $g\ge 0$, the homeomorphism type determines $g$ uniquely, and the value depends only on the topological type of $X$, not on the holomorphic atlas used to exhibit it.

The displayed formula $\chi (X)=2-2g(X)$ is the content of the in-run corollary Euler characteristic of an orientable compact surface: it states that a nonempty compact connected orientable boundaryless topological 2-manifold has a unique genus $g\ge 0$ — the sphere being $g=0$ — and Euler characteristic $2-2g$, so the alternating count is independent of the triangulation and of the polygonal schema chosen. The empty reduced word denotes the zero-handle terminal case; it is not itself a polygonal schema. The sphere has the actual one-face digon $a{a}^{-1}$, with two quotient vertices, one paired edge and one face, so $\chi =2-1+1=2$. For $g\ge 1$ the $g$-fold commutator word gives one vertex, $2g$ edges and one face, so $\chi =1-2g+1=2-2g$. The same value is obtained from any finite triangulation by the subdivision-invariance of $V-E+F$. The formula also shows that $\chi (X)$ is even and at most $2$.

Axiom of Choice. This definition assumes AC because the topological classification theorem it invokes does; AC enters exactly through Classification of compact connected surfaces and its finite triangulation and Schoenflies chain (The Axiom of Choice). No further choice is made here: the genus is read off from the classification, and the cell count is finite.

For the two basic cases: the Riemann sphere has $g=0$ and $\chi =2$, and the complex torus $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$ has $g=1$ and $\chi =0$, matching the count $1-2+1=0$ for its commutator polygon. The Euler characteristic is used in this pair only through the two identities $\chi (X)=2-2g(X)$ and $\chi (Y)=2-2g(Y)$ substituted into the cell count of Riemann–Hurwitz formula for compact Riemann surfaces; no other surface invariant is asserted here. Because the genus is defined topologically, it is automatically invariant under biholomorphism, and the notation $g(X)$ may be used before the homological or de Rham interpretations of the genus, which belong to later pages.