There exists any theorem saying that every orientable compact 2-Manifold can be embedded in $\mathbb R^3$?

Question

in my Introduction to Algebraic Topology course we defined orientable compact 2-Manifold as quotient spaces of two-dimensional shapes (so, following this definition, they appear to be just "abstract" topological spaces). Since it's possible to visualize them as subspaces of $\mathbb R^3$, i was wondering if there is a way to show that these "abstract" spaces can be embedded in $\mathbb R^3$. Thanks in advance.

Answer 1

Every compact orientable 2-manifold admits an embedding into $\mathbb R^3$. (Indeed, this turns out to be true even without the compactness assumption, see this answer).

The statement about compact 2-manifolds follows e.g. from the classification of surfaces, which is a classic result for which there are by now many proofs. It says that any compact orientable surface is homeomorphic to a surface of the form $S_g^n$, the (standard) surface of genus $g$ with $n$ boundary components, which can be explicitly embedded into $\mathbb R^3$.

A very short proof of the classification of surfaces (assuming that you know that any surface admits a triangulation, which is shown e.g. in an appendix to Hatcher's textbook on algebraic topology), due to Zeeman, is described in this note by Andrew Putnam. Alternatively, you can prove the classification (assuming smooth structures) using Morse theory.