A short proof that compact 2-manifolds can be triangulated

Abstract

The result mentioned in the title of this paper was first proved by RADO [1]; a proof can also be found in [2]. The idea for the present proof is that of the first-named author, who discovered it while investigating the possibilities of engulfing in low dimensions. It is shorter than previous proofs, and is presented in the interests of economy.We commence by listing a few familiar facts from geometric topology:The Jordan-Schoenflies Theorem. A simple closed curve J in E ~ separates E 2 into two regions. There exists a self-homeomorphism of E 2 under which J is mapped onto a circle. Thickening an Arc. Each arc in the interior of a 2-manifoM lies in the interior of a 2-cell. This 2-cell can be chosen to be disjoint from anypreassigned compact set in the complement of the are. Cellularity and Quotients. For our purposes, a cellular set K is one that can be written as the intersection of a sequence of 2-cells oo K= 0 E~, where E~cIntE~_ 1 (i=2,3 .... ). i=l If K is a cellular subset of a 2-manifoM M, then M/K is homeomorphic to M. (The corresponding statement also holds in n dimensions; see [3], for example.) We shall also have need for the following Lemma. Let M be a closed 2-manifold, and let C be a connected subset of M which is the union of n simple closed curves, C= U c i. i=1 Let A be a compact, totally-disconnected subset of C. Then .4 lies in the interior of a closed 2-cell in M. (A totally-disconnected set is characterized by the property that each of its connected subsets consists of at most one point.)Proof. Since A is compact and totally-disconnected, some subarc S of C1 contains no points of A. Thicken C1-S to obtain a closed 2-cell D1 whose interior contains A n C 1.

Jordan and Schoenflies in non-metrical analysis situs

“…By compactness we may cover the image of a contracting homotopy shrinking C to a point by a finite number of charts (open sets homeomorphic to the plane R 2 ). So C is contained in a certain Lindelöf (hence metric) subsurface M * into which C is null-homotopic 12 . By (HST) it follows that C bounds a disc in M * , which can of course be regarded as embedded in M.…”

“…sised that the Ahlfors-Sario proof stays very close to the 1925 proof of Radó. Other proofs (usually restricted to the compact case) are given in [Doyle-Moran, 1968]= [12] and [Thomassen, 1992]. In the latter reference there is (on page 116) a (too?)…”

recorrido por el concepto de curvatura

“…A toda superficie cerrada orientada S podemos asociarle un número entero χ(S) llamado la característica de Euler-Poincaré de S. Una manera de calcular a χ(S) es la siguiente. Un resultado muy importante en topología es que las superficies compactas se pueden triangular (ver (Doyle and Moran, 1968)), es decir, podemos encontrar un poliedro P que sea homeomorfo a una superficie S dada. En la Figura 17 se muestran triangulaciones de la esfera 8 y del toro 9 : José Luis Cisneros Molina Un recorrido por el concepto de curvatura Figura 17: Triangulaciones de la esfera y el toro.…”