The Classification Theorem for Connected Compact 2-Manifolds

Lemma 1: Every connected compact $2$-manifold is homeomorphic to a quotient space induced by a filled polygon with $2n$ edges with pairs of edges identified in a particular orientation.

Theorem 2(The Classification Theorem for Connected Compact 2-Manifolds): If $X$ is a connected, compact, $2$-manifold, then $X$ is homeomorphic to either the $2$-sphere, a connected sum of a finite number of tori, or a connected sum of a finite number of projective planes.

• Partial Proof: By Lemma 1 we have that every connected compact $2$-manifold is homeomorphic to a quotient space from a filled polygon with $2n$-edges where the edges are identified in pairs. We define a pair of edges to be of the first type if they have opposite orientations, and we define a pair of edges to be of the second type if they have the same orientation as illustrated below:

• We now proceed to identify all filled polygons with $2n$-edges.

• A. Identifying Polygons with $2$ Edges:

• We first identify the identifying polygons with $2$ edges. There are two of them as illustrated below. The first is homeomorphic to to the sphere, and the second is homeomorphic to the projective plane.

• The $2$-sphere can be associated with the group word $aa^{-1} = 1$, while the projective plane can be associated with the group word $a^2 = 1$.

• B. Identfying Polygons with $4 \geq$ Edges:

• Step 1 (Adjacent Edge Pairs of the First Type): If an identifying polygon has a pair of adjacent edges and they are of the first type, then we can identify them as illustrated below:

• Step 2 (Grouping Edges of the Second Type): If an identifying polygon has a pair of edges of the second type. We make a cut along the line connecting the initial points of the edges. We give this edge an orientation and a new label. We then separate the two pieces of the identifying polygon and glue them along the initial pair of edges.

• We continue this process until all pairs of edges of the second type are adjacent. We may bring edges closer if necessary with cuts and pastes of the form:

• The end result is an identifying polygon whose edges of the second type are all adjacent, and whose pairs of edges of the second type are all adjacent. The remain edges will be of the first type.

• Step 3 (Grouping Edges of the First Type): If there are any edges of the first type there will be at least four of them. We cut and paste as necessary until every quadruple of edges of the first type are of the form $aba^{-1}b^{-1}$. We continue this process.

• Step 4: We now obtain an identifying polygon where the edges of the second type are grouped and the edges of the first type are grouped in quadruples of the form $aba^{-1}b^{-1}$ to obtain a polygon that looks like:

• There are now two possibilities. Suppose that the identifying polygon contains only quadruples of edges of the first type. If there are $m$ many quadruples, then the polygon corresponds to a connected sum of $m$ tori.

• If the identifying polygon contains at least one pair of edges of the second type, then we can cut and paste to obtain an identifying polygon which only contains edges of the second type. If there are $n$ many pairs, then the polygon correspond to a connected sum of $n$ projective planes.

Corollary 3: Let $X$ be a connected compact $2$-manifold. a) If $X$ is the sphere then $\pi_1(X, x)$ is isomorphic to the trivial group. b) If $X$ is a connected sum of $m$ tori then $\pi_1(X, x) = \langle a_1, b_1, a_2, b_2, ..., a_m, b_n : a_1b_1a_1^{-1}b_1^{-1}a_2b_2a_2^{-1}b_2^{-1}...a_mb_ma_m^{-1}b_m^{-1} = 1 \rangle$. c) If $X$ is a connected sum of $n$ projective planes then $\pi_1(X, x) = \langle a_1, a_2, ..., a_n : a_1^2a_2^2...a_n^2 = 1 \rangle$.

• Proof: Apply the Seifert-van Kampen theorem to the identifying polygons.

Corollary 3 immediately gives us a way to determine if a space $X$ is connected compact $2$-manifold. If $\pi_1(X, x)$ is not isomorphic to any of the groups above, then it cannot be a connected compact $2$-manifold.