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The raw cell-count triple is a surface invariant

Statement

False: for a compact surface $M$ , the raw triple $(V, E, F)$ of vertex, edge, and face counts is independent of the chosen finite curvilinear triangulation. The alternating sum $V - E + F$ is a surface invariant by a separate comparison theorem; this false claim concerns the three counts individually.

Facts & Assumptions

Given: The assertion that the raw cell-count triple is independent of triangulation, to be refuted by two triangulations of one disk.

[F1]

A curvilinear triangulation of a compact surface is finite data $(V, E, F, ϕ)$ with $V$ vertices, $E$ edges and $F$ triangular faces (Curvilinear face-to-face triangulation).

[F2]

Inserting a vertex in an edge and splitting the incident face along a new edge, or coning a vertex inside a face to its three vertices, changes the raw counts to $(V + 1, E + 3, F + 2)$ in the interior-edge case, $(V + 1, E + 2, F + 1)$ in the boundary-edge case and $(V + 1, E + 3, F + 2)$ in the face-cone case, and always leaves $V - E + F$ unchanged (Euler count under elementary subdivision).

[F3]

The Euler characteristic of a triangulated surface is $χ (M; T) = V - E + F$ ; the number is defined for the supplied triangulation only, and independence from the choice of triangulation, and hence the notation $χ (M)$ , is not assumed in the definition (Euler characteristic of a finitely triangulated compact surface).

[F4]

Every two finite face-to-face piecewise $C^{2}$ curvilinear triangulations of a compact smooth surface have the same $V - E + F$ , and the common value is written $χ (M)$ (Topological well-definedness of the surface Euler characteristic).

Refutation

technique · exhibit two triangulations of the same disk with different vertex and edge counts, then identify the separate theorem that makes their alternating sums equal

1.1F1F2given

Let $D$ be the closed unit disk and let $T_{1}$ be its triangulation by a single curvilinear triangle, so $V_{1} = 3$ , $E_{1} = 3$ , $F_{1} = 1$ . Insert a point in the interior of one edge and split the face by the new edge from that point to the opposite vertex, obtaining a triangulation $T_{2}$ of the same surface $D$ ; this is one of the elementary subdivisions of [F2] and gives $V_{2} = 4$ , $E_{2} = 5$ , $F_{2} = 2$ . Both are curvilinear triangulations of the same compact surface by [F1], but the raw data differ: $(3, 3, 1) \neq (4, 5, 2)$ .

2.1step 1.1

The two triangulations in step 1.1 have different raw triples, $(3, 3, 1) \neq (4, 5, 2)$ , although their underlying surface is the same disk. This directly refutes the stated independence of $(V, E, F)$ .

2.2F2F3F4step 1.1algebra

The alternating sums agree, $3 - 3 + 1 = 1 = 4 - 5 + 2$ , as the subdivision lemma [F2] predicts. The definition [F3] initially writes this value as $χ (D; T)$ ; the comparison theorem [F4] proves that it is independent of $T$ and may be written $χ (D)$ . Thus this example changes the individual counts, not Euler characteristic.

3.1step 2.1step 2.2∎

Hence the raw triple $(V, E, F)$ is not a surface invariant, while the alternating sum is invariant by step 2.2.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, printed pp. 167-172, and Datar, Lectures on Riemannian Geometry, Lecture 2, Section 2.2, printed pp. 13-15, identify the alternating count of a triangulation with Euler characteristic through a comparison. The two-triangulation witness is the elementary subdivision of Euler count under elementary subdivision. The indexed definition Euler characteristic of a finitely triangulated compact surface and comparison theorem Topological well-definedness of the surface Euler characteristic explain why the alternating sum remains invariant when the raw cell counts change.

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Sources

John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997) (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)