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content/Math4202/Math4202_L29.md

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+# Math4202 Topology II (Lecture 29)
+
+## Algebraic Topology
+
+### Fundamental Groups of Some Surfaces
+
+Recall from previous lecture, we talked about figure 8 shape.
+
+#### Lemma The fundamental group of figure-8 is not abelian
+
+The fundamental group of figure-8 is not abelian.
+
+<details>
+<summary>Proof</summary>
+
+Consider $U,V$ be two "fish shape" where $U\cup V$ is the figure-8 shape, and $U\cap V$ is $x$ shape.
+
+The $x$ shape is path connected, 
+
+$\pi_1(U,x_0)$ is isomorphic to $\pi_1(S^1,x_0)$, and $\pi_1(V,x_0)$ is isomorphic to $\pi_1(S^1,x_0)$.
+
+To show that is not abelian, we need to show that $\alpha*\beta\neq \beta*\alpha$.
+
+We will use covering map to do this.
+
+[Universal covering of figure-8](https://notenexta.trance-0.com/Math4202/universal-covering-of-figure-8.png)
+
+However, for proving our result, it is sufficient to use xy axis with loops on each integer lattice.
+
+And $\tilde{\alpha*\beta}(1)=(1,0)$ and $\tilde{\beta*\alpha}(1)=(0,1)$. By path lifting correspondence, the two loops are not homotopic.
+
+</details>
+
+#### Theorem for fundamental groups of double torus (Torus with genus 2)
+
+The fundamental group of Torus with genus 2 is not abelian.
+
+<details>
+<summary>Proof</summary>
+
+If we cut the torus in the middle, we can have $U,V$ is two "punctured torus", which is homotopic to the figure-8 shape.
+
+But the is trick is not enough to show that the fundamental group is not abelian.
+
+---
+
+First we use quotient map $q_1$ to map double torus to two torus connected at one point.
+
+Then we use quotient map $q_2$ to map two torus connected at one point to figure-8 shape.
+
+So $q=q_2\circ q_1$ is a quotient map from double torus to figure-8 shape.
+
+Then consider the inclusion map $i$ and let the double torus be $X$, we claim that $i_*:\pi_1(\infty,x_0)\to \pi_1(X,x_0)$ is injective.
+
+If $\pi_1(X,x_0)$ is abelian, then the figure 8 shape is abelian, that is contradiction.
+
+</details>

content/Math4202/_meta.js

@@ -34,4 +34,5 @@ export default {
     Math4202_L26: "Topology II (Lecture 26)",
     Math4202_L27: "Topology II (Lecture 27)",
     Math4202_L28: "Topology II (Lecture 28)",
+    Math4202_L29: "Topology II (Lecture 29)",
 }

content/Math4302/_meta.js

@@ -1,5 +1,6 @@
 export default {
     index: "Course Description",
+    Exam_reviews: "Exam reviews",
     "---":{
         type: 'separator'
     },