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content/Math4202/Math4202_L10.md

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+# Math4202 Topology II (Lecture 10)
+
+## Algebraic Topology
+
+### Path homotopy
+
+
+#### Theorem for properties of product of paths
+
+1. If $f\simeq_p f_1, g\simeq_p g_1$, then $f*g\simeq_p f_1*g_1$. (Product is well-defined)
+2. $([f]*[g])*[h]=[f]*([g]*[h])$. (Associativity)
+3. Let $e_{x_0}$ be the constant path from $x_0$ to $x_0$, $e_{x_1}$ be the constant path from $x_1$ to $x_1$. Suppose $f$ is a path from $x_0$ to $x_1$.
+    $$
+    [e_{x_0}]*[f]=[f],\quad [f]*[e_{x_1}]=[f]
+    $$
+    (Right and left identity)
+4. Given $f$ in $X$ a path from $x_0$ to $x_1$, we define $\bar{f}$ to be the path from $x_1$ to $x_0$ where $\bar{f}(t)=f(1-t)$.
+    $$
+    f*\bar{f}=e_{x_0},\quad \bar{f}*f=e_{x_1}
+    $$
+    $$
+    [f]*[\bar{f}]=[e_{x_0}],\quad [\bar{f}]*[f]=[e_{x_1}]
+    $$
+
+<details>
+<summary>Proof</summary>
+
+(1) If $f\simeq_p f_1$, $g\simeq_p g_1$, then $f*g\simeq_p f_1*g_1$.
+
+Let $F$ be homotopy between $f$ and $f_1$, $G$ be homotopy between $g$ and $g_1$.
+
+We can define
+
+$$
+F*G:[0,1]\times [0,1]\to X,\quad F*G(s,t)=\left(F(-,t)*G(-,t)\right)(s)=\begin{cases}
+F(2s,t) & 0\leq s\leq \frac{1}{2}\\
+G(2s-1,t) & \frac{1}{2}\leq s\leq 1
+\end{cases}
+$$
+
+$F*G$ is a homotopy between $f*g$ and $f_1*g_1$.
+
+We can check this by enumerating the cases from definition of homotopy.
+
+---
+
+(2) $([f]*[g])*[h]=[f]*([g]*[h])$.
+
+For $f*(g*h)$, along the interval $[0,\frac{1}{2}]$ we map $x_1\to x_2$, then along the interval $[\frac{1}{2},\frac{3}{4}]$ we map $x_2\to x_3$, then along the interval $[\frac{3}{4},1]$ we map $x_3\to x_4$.
+
+For $(f*g)*h$, along the interval $[0,\frac{1}{4}]$ we map $x_1\to x_2$, then along the interval $[\frac{1}{4},\frac{1}{2}]$ we map $x_2\to x_3$, then along the interval $[\frac{1}{2},1]$ we map $x_3\to x_4$.
+
+We can construct the homotopy between $f*(g*h)$ and $(f*g)*h$ as follows.
+
+Let $f((4-2t)s)$ for $F(s,t)$,
+
+when $t=0$, $F(s,0)=f(4s)\in f*(g*h)$, when $t=1$, $F(s,1)=f(2s)\in (f*g)*h$.
+
+....
+
+_We make the linear maps between $f*(g*h)$ and $(f*g)*h$ continuous, then $f*(g*h)\simeq_p (f*g)*h$. With our homotopy constructed above_
+
+---
+
+(3) $e_{x_0}*f\simeq_p f\simeq_p f*e_{x_1}$.
+
+We can construct the homotopy between $e_{x_0}*f$ and $f$ as follows.
+
+$$
+H(s,t)=\begin{cases}
+x_0 & t\geq 2s\\
+f(2s-t) & t\leq 2s
+\end{cases}
+$$
+
+or you may induct from $f(\frac{s-t/2}{1-t/2})$ if you like.
+
+---
+
+(4) $f*\bar{f}=e_{x_0},\quad \bar{f}*f=e_{x_1}$.
+
+Note that we don't need to reach $x_1$ every time.
+
+$f_t=f(ts)$ $s\in[0,\frac{1}{2}]$.
+
+$\bar{f}_t=\bar{f}(1-ts)$ $s\in[\frac{1}{2},1]$.
+
+</details>
+
+> [!CAUTION]
+>
+> Homeomorphism does not implies homotopy automatically.
+
+#### Definition for the fundamental group
+
+The fundamental group of $X$ at $x$ is defined to be
+
+$$
+(\Pi_1(X,x),*)
+$$

content/Math4202/_meta.js

@@ -12,4 +12,5 @@ export default {
     Math4202_L7: "Topology II (Lecture 7)",
     Math4202_L8: "Topology II (Lecture 8)",
     Math4202_L9: "Topology II (Lecture 9)",
+    Math4202_L10: "Topology II (Lecture 10)",
 }