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Euler characteristic under finite subcomplex gluing

Statement

Let $A$ and $B$ be compact smooth surfaces, or compact regular surface regions in a common ambient smooth surface, each carrying a finite curvilinear triangulation in the sense of Curvilinear face-to-face triangulation, and suppose the two triangulations agree on $C = A \cap B$ : the intersection is a common finite one-dimensional subcomplex, consisting of finitely many vertices and edges shared by the two triangulations with the same incidences, so that it is a finite disjoint union of embedded compact arcs and circles and the union $A \cup B$ carries the finite curvilinear triangulation obtained by taking the union of the two cell structures. Then $χ (A \cup B) = χ (A) + χ (B) - χ (C), χ (C) = V_{C} - E_{C} .$ No claim is made that arbitrary smooth curves or arbitrary overlay intersections form such a compatible finite subcomplex.

Facts & Assumptions

Given: Compact surfaces or compact regular surface regions $A$ and $B$ with finite curvilinear triangulations agreeing on the common finite one-dimensional subcomplex $C = A \cap B$ , and the union triangulation built from the two cell structures.

[F1]

A curvilinear triangulation is finite data with vertices, regular $C^{2}$ edges and triangular faces meeting face-to-face, so finitely many such data can be combined along a common subcomplex (Curvilinear face-to-face triangulation).

[F2]

The vertices, edge interiors and face interiors of a finite curvilinear triangulation form a finite regular CW complex whose $0$ -, $1$ - and $2$ -cells are indexed by $V$ , $E$ and $F$ (A curvilinear triangulation gives a finite regular CW complex).

[F3]

For a curvilinear triangulation the Euler characteristic of the triangulated surface is $χ (M; T) = V - E + F$ , and for a compact smooth surface the common value over all triangulations is written $χ (M)$ (Euler characteristic of a finitely triangulated compact surface, Topological well-definedness of the surface Euler characteristic).

Proof

technique · count the cells of the union and apply inclusion-exclusion to the three finite counts

1.1F1given

Let $V_{A}, E_{A}, F_{A}$ and $V_{B}, E_{B}, F_{B}$ be the cell numbers of the two triangulations, and $V_{C}, E_{C}$ those of the common subcomplex $C$ . Since the triangulations agree on $C$ , the union of the two cell structures has vertex set $V_{A} \cup V_{B}$ , edge set $E_{A} \cup E_{B}$ and face set $F_{A} ⊔ F_{B}$ : a face of $A$ and a face of $B$ cannot have overlapping interiors because $A \cap B = C$ is one-dimensional, while every edge and vertex of the two triangulations lies in $A$ or in $B$ and the shared ones lie in $C$ .

2.1step 1.1algebra

Counting cell incidences of the union gives $V = V_{A} + V_{B} - V_{C}$ , $E = E_{A} + E_{B} - E_{C}$ and $F = F_{A} + F_{B}$ , since the shared vertices and edges are counted once in each triangulation and once in $C$ , and the faces have no overlap.

2.2F1F2F3step 1.1given

The union data are a finite curvilinear triangulation of $A \cup B$ : faces and edges are those of the two triangulations, they meet face-to-face because within each triangulation they do and across $C$ because the two triangulations agree on $C$ edge by edge and vertex by vertex, and the link conditions glue along the common subcomplex. Hence [F3] applies and $χ (A \cup B) = V - E + F$ , while $χ (A) = V_{A} - E_{A} + F_{A}$ and $χ (B) = V_{B} - E_{B} + F_{B}$ by the well-definedness of the Euler characteristic of compact surfaces.

3.1step 2.1step 2.2algebra

Substituting step 2.1 into the counting expression and regrouping, $V - E + F = (V_{A} - E_{A} + F_{A}) + (V_{B} - E_{B} + F_{B}) - (V_{C} - E_{C}) = χ (A) + χ (B) - χ (C)$ , where the last term is the count $χ (C) = V_{C} - E_{C}$ of the common one-dimensional subcomplex.

4.1F3step 2.2step 3.1algebra∎

Combining the two preceding steps and using the well-definedness of $χ$ for compact surfaces gives $χ (A \cup B) = χ (A) + χ (B) - χ (C)$ for every compatible gluing along a finite one-dimensional subcomplex, as claimed.

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, use the additivity of the vertex-edge-face count under pasting along a common boundary piece. The counting is carried out here directly for the union cell structure, using the CW interpretation of A curvilinear triangulation gives a finite regular CW complex and the well-defined surface Euler characteristic of Topological well-definedness of the surface Euler characteristic; the compatible-subcomplex hypothesis is stated explicitly instead of being assumed for arbitrary smooth overlays.

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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)