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content/Math4202/Math4202_L11.md

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+# Math4201 Topology II (Lecture 11)
+
+## Algebraic topology
+
+### Fundamental group
+
+The $*$ operation has the following properties:
+
+#### Properties for the path product operation
+
+Let $[f],[g]\in \Pi_1(X)$, for $[f]\in \Pi_1(X)$, let $s:\Pi_1(X)\to X, [f]\mapsto f(0)$ and $t:\Pi_1(X)\to X, [f]\mapsto f(1)$.
+
+Note that $t([f])=s([g])$, $[f]*[g]=[f*g]\in \Pi_1(X)$.
+
+This also satisfies the associativity. $([f]*[g])*[h]=[f]*([g]*[h])$.
+
+We have left and right identity. $[f]*[e_{t(f)}]=[f], [e_{s(f)}]*[f]=[f]$.
+
+We have inverse. $[f]*[\bar{x}]=[e_{s(f)}], [\bar{x}]*[f]=[e_{t(f)}]$
+
+#### Definition for Groupoid
+
+Let $f,g$ be paths where $g,f:[0,1]\to X$, and consider the function of all pathes in $G$, denoted as $\mathcal{G}$,
+
+Set $t:\mathcal{G}\to X$ be the source map, for this case $t(f)=f(0)$, and $s:\mathcal{G}\to X$ be the target map, for this case $s(f)=f(1)$.
+
+We define
+
+$$
+\mathcal{G}^{(2)}=\{(f,g)\in \mathcal{G}\times \mathcal{G}|t(f)=s(g)\}
+$$
+
+And we define the operation $*$ on $\mathcal{G}^{(2)}$ as the path product.
+
+This satisfies the following properties:
+
+- Associativity: $(f*g)*h=f*(g*h)$
+
+Consider the function $\eta:X\to \mathcal{G}$, for this case $\eta(x)=e_{x}$.
+
+- We have left and right identity: $\eta(t(f))*f=f, f*\eta(s(f))=f$
+
+- Inverse: $\forall g\in \mathcal{G}, \exists g^{-1}\in \mathcal{G}, g*g^{-1}=\eta(s(g))$, $g^{-1}*g=\eta(t(g))$
+
+#### Definition for loop
+
+Let $x_0\in X$. A path starting and ending at $x_0$ is called a loop based at $x_0$.
+
+#### Definition for the fundamental group
+
+The fundamental group of $X$ at $x$ is defined to be
+
+$$
+(\Pi_1(X,x),*)
+$$
+
+where $*$ is the product operation, and $\Pi_1(X,x)$ is the set o homotopy classes of loops in $X$ based at $x$.
+
+<details>
+<summary>Example of fundamental group</summary>
+
+Consider $X=[0,1]$, with subspace topology from standard topology in $\mathbb{R}$.
+
+$\Pi_1(X,0)=\{e\}$, (constant function at $0$) since we can build homotopy for all loops based at $0$ as follows $H(s,t)=(1-t)f(s)+t$.
+
+And $\Pi_1(X,1)=\{e\}$, (constant function at $1$.)
+
+---
+
+Let $X=\{1,2\}$ with discrete topology.
+
+$\Pi_1(X,1)=\{e\}$, (constant function at $1$.)
+
+$\Pi_1(X,2)=\{e\}$, (constant function at $2$.)
+
+---
+
+Let $X=S^1$ be the circle.
+
+$\Pi_1(X,1)=\mathbb{Z}$ (related to winding numbers, prove next week).
+
+</details>
+
+A natural question is, will the fundamental group depends on the basepoint $x$?
+
+#### Definition for $\hat{\alpha}$
+
+Let $\alpha$ be a path in $X$ from $x_0$ to $x_1$. $\alpha:[0,1]\to X$ such that $\alpha(0)=x_0$ and $\alpha(1)=x_1$. Define $\hat{\alpha}:\Pi_1(X,x_0)\to \Pi_1(X,x_1)$ as follows:
+
+$$
+\hat{\alpha}(\beta)=[\bar{\alpha}]*[f]*[\alpha]
+$$
+
+#### $\hat{\alpha}$ is a group homomorphism
+
+$\hat{\alpha}$ is a group homomorphism between $(\Pi_1(X,x_0),*)$ and $(\Pi_1(X,x_1),*)$
+
+<details>
+<summary>Proof</summary>
+
+Let $f,g\in \Pi_1(X,x_0)$, then $\hat{\alpha}(f*g)=\hat{\alpha}(f)\hat{\alpha}(g)$
+
+$$
+\begin{aligned}
+\hat{\alpha}(f*g)&=[\bar{\alpha}]*[f]*[g]*[\alpha]\\
+&=[\bar{\alpha}]*[f]*[e_{x_0}]*[g]*[\alpha]\\
+&=[\bar{\alpha}]*[f]*[\alpha]*[\bar{\alpha}]*[g]*[\alpha]\\
+&=([\bar{\alpha}]*[f]*[\alpha])*([\bar{\alpha}]*[g]*[\alpha])\\
+&=(\hat{\alpha}(f))*(\hat{\alpha}(g))
+\end{aligned}
+$$
+
+---
+
+Next, we will show that $\hat{\alpha}\circ \hat{\bar{\alpha}}([f])=[f]$, and $\hat{\bar{\alpha}}\circ \hat{\alpha}([f])=[f]$.
+
+$$
+\begin{aligned}
+\hat{\alpha}\circ \hat{\bar{\alpha}}([f])&=\hat{\alpha}([\bar{\alpha}]*[f]*[\alpha])\\
+&=[\alpha]*[\bar{\alpha}]*[f]*[\alpha]*[\bar{\alpha}]\\
+&=[e_{x_0}]*[f]*[e_{x_1}]\\
+&=[f]
+\end{aligned}
+$$
+
+The other case is the same
+
+</details>
+
+#### Corollary of fundamental group
+
+If $X$ is path-connected and $x_0,x_1\in X$, then $\Pi_1(X,x_0)$ is isomorphic to $\Pi_1(X,x_1)$.

content/Math4202/_meta.js

@@ -13,4 +13,5 @@ export default {
     Math4202_L8: "Topology II (Lecture 8)",
     Math4202_L9: "Topology II (Lecture 9)",
     Math4202_L10: "Topology II (Lecture 10)",
+    Math4202_L11: "Topology II (Lecture 11)",
 }

content/Math4302/Math4302_L10.md

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-# Math4302 Modern Algebra (Lecture 9)
+# Math4302 Modern Algebra (Lecture 10)
 
 ## Groups
 

content/Math4302/Math4302_L11.md

@@ -0,0 +1 @@
+# Math4302 Modern Algebra (Lecture 11)

content/Math4302/_meta.js

@@ -13,4 +13,5 @@ export default {
     Math4302_L8: "Modern Algebra (Lecture 8)",
     Math4302_L9: "Modern Algebra (Lecture 9)",
     Math4302_L10: "Modern Algebra (Lecture 10)",
+    Math4302_L11: "Modern Algebra (Lecture 11)",
 }