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Take the proof that any compact smooth manifold admits triangulations, and set the dimension to two.

The idea goes like this:

• Embed your surface (or $n$-manifold) in $\mathbb R^5$ ($\mathbb R^{2n+1}$ in general).

• Triangulate $\mathbb R^5$, and make the surface transverse to the triangulation. If the surface does not intersect each simplex in a locally linear manner, subdivide the triangulation and repeat this step until it does.

• The pull-back of the triangulation to the surface is a decomposition into convex polyhedra. A subdivision turns this into a triangulation.

Paraphrasing Allen Hatcher:

If you're interested in topological surfaces, the paper

A.J.S. Hamilton, The triangulation of 3-manifolds, Oxford Quart. J. Math. 27 (1976), 63-70

takes the Kirby-Siebenmann machinery and scales it down to 3 dimensions where it becomes somewhat simpler, so one can prove existence and uniqueness of triangulations of 3-manifolds using only standard PL techniques, such as results of Waldhausen. Presumably the same approach would work for surfaces. Since the method works in 3 dimensions it can't be using the topological Shoenflies theorem since this fails in 3 dimensions. On the other hand, it would use some PL (or smooth) surface theory so it wouldn't be entirely "from scratch".

edit: Allen wrote this argument up in a recent paper. See this thread for details http://mathoverflow.net/a/151760/353

Take the proof that any compact smooth manifold admits triangulations, and set the dimension to two.

The idea goes like this:

• Embed your surface (or $n$-manifold) in $\mathbb R^5$ ($\mathbb R^{2n+1}$ in general).

• Triangulate $\mathbb R^5$, and make the surface transverse to the triangulation. If the surface does not intersect each simplex in a locally linear manner, subdivide the triangulation and repeat this step until it does.

• The pull-back of the triangulation to the surface is a decomposition into convex polyhedra. A subdivision turns this into a triangulation.

Paraphrasing Allen Hatcher:

If you're interested in topological surfaces, the paper

A.J.S. Hamilton, The triangulation of 3-manifolds, Oxford Quart. J. Math. 27 (1976), 63-70

takes the Kirby-Siebenmann machinery and scales it down to 3 dimensions where it becomes somewhat simpler, so one can prove existence and uniqueness of triangulations of 3-manifolds using only standard PL techniques, such as results of Waldhausen. Presumably the same approach would work for surfaces. Since the method works in 3 dimensions it can't be using the topological Shoenflies theorem since this fails in 3 dimensions. On the other hand, it would use some PL (or smooth) surface theory so it wouldn't be entirely "from scratch".

edit: Allen wrote this argument up in a recent paper. See this thread for details https://mathoverflow.net/a/151760/353

Take the proof that any compact smooth manifold admits triangulations, and set the dimension to two.

The idea goes like this:

• Embed your surface (or $n$-manifold) in $\mathbb R^5$ ($\mathbb R^{2n+1}$ in general).

• Triangulate $\mathbb R^5$, and make the surface transverse to the triangulation. If the surface does not intersect each simplex in a locally linear manner, subdivide the triangulation and repeat this step until it does.

• The pull-back of the triangulation to the surface is a decomposition into convex polyhedra. A subdivision turns this into a triangulation.

Paraphrasing Allen Hatcher:

If you're interested in topological surfaces, the paper

A.J.S. Hamilton, The triangulation of 3-manifolds, Oxford Quart. J. Math. 27 (1976), 63-70

takes the Kirby-Siebenmann machinery and scales it down to 3 dimensions where it becomes somewhat simpler, so one can prove existence and uniqueness of triangulations of 3-manifolds using only standard PL techniques, such as results of Waldhausen. Presumably the same approach would work for surfaces. Since the method works in 3 dimensions it can't be using the topological Shoenflies theorem since this fails in 3 dimensions. On the other hand, it would use some PL (or smooth) surface theory so it wouldn't be entirely "from scratch".

Take the proof that any compact smooth manifold admits triangulations, and set the dimension to two.

The idea goes like this:

• Embed your surface (or $n$-manifold) in $\mathbb R^5$ ($\mathbb R^{2n+1}$ in general).

• Triangulate $\mathbb R^5$, and make the surface transverse to the triangulation. If the surface does not intersect each simplex in a locally linear manner, subdivide the triangulation and repeat this step until it does.

• The pull-back of the triangulation to the surface is a decomposition into convex polyhedra. A subdivision turns this into a triangulation.

Paraphrasing Allen Hatcher:

If you're interested in topological surfaces, the paper

A.J.S. Hamilton, The triangulation of 3-manifolds, Oxford Quart. J. Math. 27 (1976), 63-70

takes the Kirby-Siebenmann machinery and scales it down to 3 dimensions where it becomes somewhat simpler, so one can prove existence and uniqueness of triangulations of 3-manifolds using only standard PL techniques, such as results of Waldhausen. Presumably the same approach would work for surfaces. Since the method works in 3 dimensions it can't be using the topological Shoenflies theorem since this fails in 3 dimensions. On the other hand, it would use some PL (or smooth) surface theory so it wouldn't be entirely "from scratch".

edit: Allen wrote this argument up in a recent paper. See this thread for details http://mathoverflow.net/a/151760/353

Take the proof that any compact smooth manifold admits triangulations, and set the dimension to two.

The idea goes like this:

• Embed your surface (or $n$-manifold) in $\mathbb R^5$ ($\mathbb R^{2n+1}$ in general).

• Triangulate $\mathbb R^5$, and make the surface transverse to the triangulation. If the surface does not intersect each simplex in a locally linear manner, subdivide the triangulation and repeat this step until it does.

• The pull-back of the triangulation to the surface is a decomposition into convex polyhedra. A subdivision turns this into a triangulation.

Take the proof that any compact smooth manifold admits triangulations, and set the dimension to two.

The idea goes like this:

• Embed your surface (or $n$-manifold) in $\mathbb R^5$ ($\mathbb R^{2n+1}$ in general).

• Triangulate $\mathbb R^5$, and make the surface transverse to the triangulation. If the surface does not intersect each simplex in a locally linear manner, subdivide the triangulation and repeat this step until it does.

• The pull-back of the triangulation to the surface is a decomposition into convex polyhedra. A subdivision turns this into a triangulation.

Paraphrasing Allen Hatcher:

If you're interested in topological surfaces, the paper

A.J.S. Hamilton, The triangulation of 3-manifolds, Oxford Quart. J. Math. 27 (1976), 63-70

takes the Kirby-Siebenmann machinery and scales it down to 3 dimensions where it becomes somewhat simpler, so one can prove existence and uniqueness of triangulations of 3-manifolds using only standard PL techniques, such as results of Waldhausen. Presumably the same approach would work for surfaces. Since the method works in 3 dimensions it can't be using the topological Shoenflies theorem since this fails in 3 dimensions. On the other hand, it would use some PL (or smooth) surface theory so it wouldn't be entirely "from scratch".