Topological classification of compact Riemann surfaces

($X$ is a compact connected boundaryless topological 2-manifold.) By [F1] the charts of $X$ are homeomorphisms onto open subsets of $\mathbb{C}\cong {\mathbb{R}}^2$, so $X$ is a topological 2-manifold; it has no boundary because charts take values in open sets of ${\mathbb{R}}^2$, it is nonempty and connected by the Riemann-surface convention, and it is compact by hypothesis.

(The holomorphic atlas orients $X$.) For charts $\phi ,\psi$ with transition $\tau =\psi \circ {\phi }^{-1}$ on a plane domain, $\tau$ is biholomorphic onto its image, hence injective and holomorphic, so ${\tau }^{\mathrm{\prime }}\mathrm{\ne }0$ everywhere by [F2] and then $\det J\tau =\mathrm{\mid }{\tau }^{\mathrm{\prime }}{\mathrm{\mid }}^2>0$ everywhere by [F2]. Transporting the standard orientation of $\mathbb{C}$ through each chart therefore gives local orientations that agree on every overlap; equivalently, the maximal atlas is an oriented atlas. The resulting local orientation data are canonical for the holomorphic structure: any chart compatible with the atlas has biholomorphic transitions to the charts of the atlas, hence the same Jacobian positivity, so it defines the same orientation.

(Finite chart triangulation.) Applying [F5] with $Y=X$ and $F=\mathrm{\varnothing }$ gives a finite oriented topological triangulation of $X$ by chart-contained triangles; this is the finite triangulation on which the polygonal reduction of [F4] operates.

(The atlas orientation is the integral orientability read by [F4].) The orientation of step 1.2 gives, at every $x\in X$ and in every chart around $x$, the generator of the local homology $H_2(X,X\setminus \{x\};\mathbb{Z})$ determined by the standard orientation of the plane through that chart; the transition computation of step 1.2 shows the generator is independent of the chart, and it varies continuously because it is locally induced by one chart. Hence $X$ carries an integral orientation in the sense of [F3], so $X$ is orientable, and this orientation is canonical for the holomorphic structure.

(Classification and exclusion of the nonorientable models.) By [F4], whose hypothesis is satisfied by the compact connected boundaryless 2-manifold $X$ of step 1.1, the surface $X$ is homeomorphic to $S^2$, to the connected sum of $g\ge 1$ tori, or to the connected sum of $k\ge 1$ projective planes, these being the canonical polygon words of the classification; and by the same theorem the models are distinguished by integral orientability and Euler characteristic. By step 2.1 the surface $X$ is orientable, and the nonorientable models are exactly the $k$-fold connected sums of the real projective plane, $k\ge 1$; identifying the square-word family with the nonorientable models is an explicit obligation on Classification of compact connected surfaces, which must deliver it together with the normal forms. Hence $X\cong S^2$ or $X\cong {\mathrm{\#}}_g{T}^2$ for some $g\ge 1$.

(Unique handle number.) The alternative $X\cong S^2$ is the case $g=0$ of $X\cong {\mathrm{\#}}_g{T}^2$, and for $g\ge 1$ the number $g$ is unique by the uniqueness clause of [F4]; hence there is exactly one $g\ge 0$ with $X\cong {\mathrm{\#}}_g{T}^2$, the sphere with $g$ handles. The chart triangulation of step 1.3 witnesses the finite triangulation fed into the polygonal reduction, and the Axiom of Choice is used exactly through [F4] by [F6]; the chart orientation of steps 1.2–1.3, the local triangulation of step 1.3 and the uniqueness conclusion use no choice principle.