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content/Math4202/Math4202_L22.md

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+# Math4202 Topology II (Lecture 22)
+
+## Final reading, report, presentation
+
+- Mar 30: Reading topic send email or discuss in OH.
+- Apr 3: Finalize the plan.
+- Apr 22,24: Last two lectures: 10 minutes to present.
+- Final: type a short report, 2-5 pages.
+
+## Algebraic topology
+
+### Fundamental theorem of Algebra
+
+For arbitrary polynomial $f(z)=\sum_{i=0}^n a_i x^i$. Are there roots in $\mathbb{C}$?
+
+Consider $f(z)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0$ is a continuous map from $\mathbb{C}\to\mathbb{C}$.
+
+If $f(z_0)=0$, then $z_0$ is a root.
+
+By contradiction, Then $f:\mathbb{C}\to\mathbb{C}-\{0\}\cong \mathbb{R}^2-\{(0,0)\}$.
+
+#### Theorem for existence of n roots
+
+A polynomial equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0=0$ of degree $>0$ with complex coefficients has at least one complex root.
+
+There are $n$ roots by induction.
+
+#### Lemma
+
+If $g:S^1\to \mathbb{R}^2-\{(0,0)\}$ is the map $g(z)=z^n$, then $g$ is not nulhomotopic. $n\neq 0$, $n\in \mathbb{Z}$.
+
+> Recall that we proved that $g(z)=z$ is not nulhomotopic.
+
+Consider $k:S^1\to S^1$ by $k(z)=z^n$. $k$ is continuous, $k_*:\pi_1(S^1,1)\to \pi_1(S^1,1)$.
+
+Where $\pi_1(S^1,1)\cong \mathbb{Z}$.
+
+$k_*(n)=nk_*(1)$.
+
+Recall that the path in the loop $p:I\to S^1$ where $p:t\mapsto e^{2\pi it}$.
+
+$k_*(p)=[k(p(t))]$, where $n=\tilde{k\circ p}(1)$.
+
+$k_*$ is injective.
+

content/Math4202/_meta.js

@@ -25,4 +25,5 @@ export default {
     Math4202_L19: "Topology II (Lecture 19)",
     Math4202_L20: "Topology II (Lecture 20)",
     Math4202_L21: "Topology II (Lecture 21)",
+    Math4202_L22: "Topology II (Lecture 22)",
 }