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.