Euler characteristic of a finitely triangulated compact surface

Let $M$ be a compact smooth two-manifold, possibly disconnected, nonorientable or with boundary, and let $\mathcal{T}=(V,E,F,\varphi )$ be a curvilinear face-to-face triangulation of $M$ in the sense of Curvilinear face-to-face triangulation. Write $V(\mathcal{T})=\mathrm{\mid }V\mathrm{\mid },E(\mathcal{T})=\mathrm{\mid }E\mathrm{\mid },F(\mathcal{T})=\mathrm{\mid }F\mathrm{\mid },$ counting each vertex once, each edge once (a boundary edge with one incident face is still a single edge), and each closed triangular face once. The Euler characteristic of the triangulated surface $(M;\mathcal{T})$ is $\chi (M;\mathcal{T}):=V(\mathcal{T})-E(\mathcal{T})+F(\mathcal{T})\mathrm{.}$

The number is defined for the supplied triangulation only. Under the axiom of choice, existence of at least one such triangulation for every compact smooth surface carrying a Riemannian metric follows from Finite curvilinear triangulation of a compact Riemannian surface. The independence of $\chi (M;\mathcal{T})$ from the choice of triangulation, and hence the notation $\chi (M)$, is not assumed here; it is proved later by the well-definedness theorem on this page. A subdivision that inserts vertices in edges or cones a vertex inside a face does not change the value, as recorded by the subdivision lemma on this page.

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, printed pp. 167-172, counts vertices, edges and faces of a geodesic triangulation in the proof of Theorem 9.7 and in Problem 9-5; Datar, Lectures on Riemannian Geometry, Lecture 2, Section 2.2, printed pp. 13-15, uses the same finite count. The definition above is indexed by the supplied triangulation, and the invariance statement is deliberately deferred to the later theorem of this page rather than imported from the sources.