Math4022 Topology II (Lecture 35)

Algebraic Topology

Classification of surfaces

Recall from previous lecture, a surface is a compact 2 dimensional manifold without boundary.

Theorem Glueing polygonal regions is compact Hausdorff space

Let $X$ be the space obtained from a finite collection of polygonal region by pasting edges together according to same labeling $X$ is a compact Hausdorff space.

Proof

Let $P_1,P_2,\cdots,P_n$ be the collection of closed subset of $S_i$ where each $S_i$ is a polygonal region that is compact, and their finite union is compact.

Since the quotient map is continuous, the image of $P_1\sqcup P_2\cup\cdots\cup P_n$ under the quotient map is compact.

It suffices to show that the quotient map is closed.

Tip

A lemma is hidden here and we don’t have it in our notes.

$C\subseteq P_1\sqcup P_2\cup\cdots\cup P_n$ is closed, and $q(C)\subseteq X$ is closed.

We need to show that $q^{-1}(q(C))$ is closed.

q^{-1}(q(C^\circ \cup \partial C))=C^\circ \cup \partial C=\{p\in \partial P_i|p\sim\in \partial C,p\notin \partial C\}

Note that for each $\partial P_1=\bigcup_{i_1}e_1^{i_1}$ , $\partial P_2=\bigcup_{i_2}e_2^{i_2}$ , $\partial P_n=\bigcup_{i_n}e_n^{i_n}$ , and so on.

And

\partial C=\bigcup_{i_1}(\partial C\cap e_1^{i_1})\cup \bigcup_{i_2}(\partial C\cap e_2^{i_2})\cup \cdots \cup \bigcup_{i_n}(\partial C\cap e_n^{i_n})

This is a finite union of closed sets, therefore closed.

Each extra piece in $q^{-1}(q(C))$ not in $C$ of ( $\partial C\cap e_n^{i_n}$ ) is homeomorphic to one of them. And $q^{-1}(q(C))$ is closed.

Theorem the quotient space by gluing is a manifold

This part is intentionally skipped, proving the theorem involve glue the vertices and show it is safe to have a local homeomorphism region. And is complicated and not in notes.

Theorem Fundamental group from the orientation

Let $P$ be a polygonal region, let

\omega=a_{i_1}^{\epsilon_1}a_{i_2}^{\epsilon_2}\cdots a_{i_n}^{\epsilon_n}

Fix a global orientation of the boundary. Let $X$ be te quotient space and $\pi:P\to X$ be the quotient map.

If $\pi$ maps all vertices to a single point $x_0$ of $X$ , and if $a_1,\cdots,a_n$ are the distinct labels appearing in the labelling scheme. Then $\pi_1(X,x_0)$ is isomorphic the quotient of the free group generated by $a_1,\cdots,a_n$ by the smallest normal subgroup containing $a_{i_1}^{\epsilon_1},a_{i_2}^{\epsilon_2},\cdots,a_{i_n}^{\epsilon_n}$ .

Example

The torus will be $a_1 a_2 a_1^{-1} a_2^{-1}$ , so the quotient space is the torus.