Classification of Compact Two-Manifolds

Theorem
Any smooth, compact, path-connected 2-manifold is diffeomorphic to $\mathbb S^2$, a connected sum of $\mathbb T^2$s, or a connected sum of $\mathbb{RP}^2$s.

Any such 2-manifold with boundary is diffeomorphic to $\mathbb S^2$, a connected sum of $\mathbb T^2$s, or a connected sum of $\mathbb{RP}^2$s, with a number of open disks removed.

The Euler characteristic, orientability, and number of boundary curves suffice to describe a surface.

Proof
It is known that the connected sum of g tori, $\mathbb T_1^2 \# \mathbb T_2^2 \# \ldots \# \mathbb T_g^2$, which we denote $g\mathbb T^2$ is orientable and has Euler characteristic 2-2g-b, where b is the number of boundary curves. It is also known that $p\mathbb{RP}^2$ is non-orientable and has Euler characteristic 2-p-b. Thus, the Euler characteristic, number of boundary curves, and orientability distinguish any closed, path connected 2-manifold.


 * Lemma: A compact, boundaryless 2-manifold S is diffeomorphic to a polyhedral disk P with edges identified pairwise; that is, for any closed, connected 2-manifold, $\exists$ a polyhedral disk P and an equivalence relation ~ such that $S \cong P \setminus$~.


 * Proof: In progress

With this lemma, the classification can be completed. Throughout the proof, we use "surface" in place of "smooth 2-manifold."

Consider the polygonal representation guaranteed to exist by the lemma above. Suppose there is only one pair of edges on P; then they are either identified in an orientable or non-orientable manner, yielding $\mathbb S^2$ or $\mathbb{RP}^2$, respectively.

Now suppose there are more than one pair of edges in P identified by ~. If it can be shown that such a surface can always be decomposed into the connected some of either a $\mathbb T^2$ or a $\mathbb{RP}^2$ and a surface described by a polyhedral disk with fewer edges than P, the classification of path-connected surface will be complete. There are 5 cases to be examined.


 * Case 1

There are two adjacent edges identified with opposite orientations. Then the identification can be performed to obtain a new polyhedral disk with one less pair of edges.


 * Case 2

There are two adjacent edges identified with the same orientations. Then there is a curve in P from the unshared points of these two edges which, under ~, becomes an simple closed curve in S.  The triangular disk $\Delta$ created by this curve and the two edges is just two edges identified with the same orientations, with a boundary. Hence $\Delta \cong \mathbb{RP}^2-\mathbb D_\Delta^2$, and therefore if we remove the two edges in question from P, we construct a polygonal disk P-$\Delta$ such that $((P-\Delta)\setminus$~$) \# \mathbb{RP}^2 \cong S$.


 * Case 3

There are two non-adjacent edges in P, identified with the same orientations. There exists a curve C in P from the endpoint of one of these edges to the identified point in the other. A new polygonal disk, and a new equivalence relation, can be defined as follows: Let P' be the identification of the edges in question and the separation of the disk along C, and let ~' be all the equivalences of ~ together with the new equivalence taking one of the copies of C in P' to the other, with appropriate orientation. P' and ~' are constructed so that $P' \setminus$~'$\cong S$, but now P' satisfies Case 2 because of the orientations imposed on the copies of C.


 * Case 4

There are two non-adjacent edges in P, identified with opposite orientation, such that neither edge is between any other pair of identified edges on the perimeter of P. If we identify the two edges in question, a cylinder Y is obtained. There is a curve C in Y such that if Y is separated along C, two cylinders are obtained. If the separation is maintained, but the remaining identifications are performed, then two surfaces, each with boundary C, are obtained, and so the original surface S was the connected sum of two surfaces, each with fewer edges in their respective polyhedral disks than S had.


 * Case 5

There are two non-adjacent edges identified with opposing orientations, such that some other pair of identified edges are interlaced with the edges in question on the perimeter of P. By the preceding cases, it is possible to decompose this surface through the removal of all edges with the same orientation, and forming a connected sum with a number of projective planes; hence we regard ~ as identifying any two edges with opposing orientations only. Let the pairs of identified edges that are interlaced be $a_1, a_2, b_1, b_2$, such that $a_1$~$a_2$ and $b_1$~$b_2$. By performing the identification on a, a cylinder is obtained, and by further performing the identification on b, the remaining edges of P form the boundary of a disk on a torus $\mathbb T^2$. Hence the original surface was the connected sum $\mathbb T^2 \# (P-a_1-a_2-b_1-b_2) \setminus$~.

Any polyhedral disk with more than one pair of identified edges must satisfy at least one of the above 5 cases, and so can be decomposed into the connected sum of either a projective plane or a torus with a surface described by a polyhedral disk with fewer pairs of identified edges. Since the polyhedral disk with only one pair of identified edges is either a sphere or a projective plane, every surface without boundary S is either a sphere or the connected sum of a collection of tori and projective planes.

Since $\mathbb T^2 \# \mathbb{RP}^2 = \mathbb{RP}^2 \# \mathbb{RP}^2 \# \mathbb{RP}^2$, any compact surface without boundary is diffeomorphic to either $\mathbb S^2, g\mathbb T^2$, or $p\mathbb{RP}^2$.

The case for surfaces with boundary is obtained through the removal and insertion of $\mathbb D^2$s to the surface.