From b076231a73a69ae32e966006379b85c8b6b3104e Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Wed, 25 Feb 2026 14:03:20 -0600 Subject: [PATCH] updates --- content/Math4202/Math4202_L19.md | 75 ++++++++++++++++++++++++++++++++ content/Math4202/_meta.js | 2 + content/Math4302/Math4302_L18.md | 75 ++++++++++++++++++++++++++++++++ content/Math4302/_meta.js | 1 + 4 files changed, 153 insertions(+) create mode 100644 content/Math4202/Math4202_L19.md create mode 100644 content/Math4302/Math4302_L18.md diff --git a/content/Math4202/Math4202_L19.md b/content/Math4202/Math4202_L19.md new file mode 100644 index 0000000..a9d3c4a --- /dev/null +++ b/content/Math4202/Math4202_L19.md @@ -0,0 +1,75 @@ +# Math4202 Topology II (Lecture 19) + +## Exam announcement + +Cover from first lecture to the fundamental group of circle. + +## Algebraic Topology + +### Retraction and fixed point + +#### Definition of retraction + +If $A\subseteq X$, a retraction of $X$ onto $A$ is a continuous map $r:X\to A$ such that $r|_A$ is the identity map of $A$. + +When such a retraction $r$ exists, $A$ is called a retract of $X$. + +
+Example + +Identity map is a retraction of $X$ onto $X$. + +--- + +$X=\mathbb{R}^2$, $A=\{0\}$, the constant map that maps all points to $(0,0)$ is a retraction of $X$ onto $A$. + +This can be generalized to any topological space, take $A$ as any one point set in $X$. + +--- + +Let $X=\mathbb{R}^2$, $A=\mathbb{R}$, the projection map that maps all points to the first coordinate is a retraction of $X$ onto $A$. + +--- + +> Can we retract $\mathbb{R}^2$ to a circle? + +Let $\mathbb{R}^2\to S^1$ + +This can be done in punctured plane. $\mathbb{R}^2\setminus\{0\}\to S^1$. by $\vec{x}\mapsto \vec{x}/\|x\|$. + +But + +
+ +#### Lemma for retraction + +If $A$ is a retract of $X$, the homomorphism of fundamental groups induced by the inclusion map $j:A\to X$, with induced $j_*:\pi_1(A,x_0)\to \pi_1(X,x_0)$ is injective. + +
+Proof + +Let $r:X\to A$ be a retraction. Consider $j:A\to X, r:X\to A$. Then $r\circ j(a)=r(a)=a$. Therefore $r\circ j=Id_A$. + +Then $r_*\circ j_*=Id_{\pi_1(A,x_0)}$. + +$\forall f\in \ker j_*$, $j_*f=0$. $r_*\circ j_*f=Id_{f}=f$, therefore $f=0$. + +So $\ker j_*=\{0\}$. + +So it is injective. +
+ +Consider the $\mathbb{R}^2\to S^1$ example, if such retraction exists, $j_*:\pi_1(S^1,x_0)\to \pi_1(\mathbb{R}^2,x_0)$ is injective. But the fundamental group of circle is $\mathbb{Z}$ whereas the fundamental group of plane is $1$. That cannot be injective. + +#### Corollary for lemma of retraction + +There is no retraction from $\mathbb{R}^2$, $B_1(0)\subseteq \mathbb{R}^2$ (unit ball in $\mathbb{R}^2$), to $S^1$. + +#### Lemma + +Let $h:S^1\to X$ be a continuous map. The following are equivalent: + +- $h$ is null-homotopic ($h$ is homotopic to a constant map). +- $h$ extends to a continuous map from $B_1(0)\to X$. +- $h_*$ is the trivial group homomorphism of fundamental groups (Image of $\pi_1(S^1,x_0)\to \pi_1(X,x_0)$ is trivial group, identity). + diff --git a/content/Math4202/_meta.js b/content/Math4202/_meta.js index 0fc3be9..be7b91d 100644 --- a/content/Math4202/_meta.js +++ b/content/Math4202/_meta.js @@ -20,4 +20,6 @@ export default { Math4202_L15: "Topology II (Lecture 15)", Math4202_L16: "Topology II (Lecture 16)", Math4202_L17: "Topology II (Lecture 17)", + Math4202_L18: "Topology II (Lecture 18)", + Math4202_L19: "Topology II (Lecture 19)", } diff --git a/content/Math4302/Math4302_L18.md b/content/Math4302/Math4302_L18.md new file mode 100644 index 0000000..4774891 --- /dev/null +++ b/content/Math4302/Math4302_L18.md @@ -0,0 +1,75 @@ +# Math4302 Modern Algebra (Lecture 18) + +## Groups + +### Factor group + +Suppose $G$ is a group, and $H\trianglelefteq G$, then $G/H$ is a group. + +Recall from last lecture, if $\phi:G\to G'$ is a homomorphism, then $G/\ker(\phi)\simeq \phi(G)\leq G'$. + +
+Example (continue from last lecture) + +$\mathbb{Z}\times\mathbb{Z}/\langle (1,1)\rangle\simeq \mathbb{Z}$ + +Take $\phi(a,b)=a-b$, this is a surjective homomorphism from $\mathbb{Z}\times\mathbb{Z}/\langle (1,1)\rangle$ to $\mathbb{Z}$ + +--- + +$\mathbb{Z}\times\mathbb{Z}/\langle (2,1)\rangle\simeq \mathbb{Z}$ + +where $\langle (2,1)\rangle=\{(2b,b)|b\in \mathbb{Z}\}$ + +Take $\phi(a,b)=a-2b$, this is a surjective homomorphism from $\mathbb{Z}\times\mathbb{Z}/\langle (2,1)\rangle$ to $\mathbb{Z}$ + +--- + +$\mathbb{Z}\times\mathbb{Z}/\langle (2,2)\rangle$ + +This should also be a finitely generated abelian group. ($\mathbb{Z}_2\times \mathbb{Z}$ actually) + +Take $\phi(a,b)=(a\mod 2,a-b)$ + +--- + +More generally, for $\mathbb{Z}\times \mathbb{Z}/\langle (a,b)\rangle$. + +This should be $\mathbb{Z}\times \mathbb{Z}_{\operatorname{gcd}(a,b)}$ + +Try to do section by gcd. + +
+ +> - If $G$ is abelian, $N\leq G$, then $G/N$ is abelian. +> - If $G$ is finitely generated and $N\trianglelefteq G$, then $G/N$ is finitely generated. + +#### Definition of simple group + +$G$ is simple if $G$ has no proper ($H\neq G,\{e\}$), normal subgroup. + +> [!TIP] +> +> In general $S_n$ is not simple, consider the normal subgroup $A_n$. + +
+Example of some natural normal subgroups + +If $\phi:G\to G'$ is a homomorphism, then $\ker(\phi)\trianglelefteq G$. + +--- + +The **center** of $G$: $Z(G)=\{a\in G|ag=ga\text{ for all }g\in G\}$ + +$Z(G)\trianglelefteq G$. + +- $e\in Z(G)$. +- $a,b\in Z(G)\implies abg=gab\implies ab\in Z(G)$. +- $a\in Z(G)\implies ag=ga\implies a^{-1}\in Z(G)$. +- If $g\in G, h\in Z(G)$, then $ghg^{-1}\in Z(G)$ since $ghg^{-1}=gg^{-1}h=h$. + +$Z(S_3)=\{e\}$, all the transpositions are not commutative, so $Z(S_3)=\{e\}$. + +$Z(GL_n(\mathbb{R}))$? continue on friday. + +
diff --git a/content/Math4302/_meta.js b/content/Math4302/_meta.js index c5cc85d..baf2b7b 100644 --- a/content/Math4302/_meta.js +++ b/content/Math4302/_meta.js @@ -20,4 +20,5 @@ export default { Math4302_L15: "Modern Algebra (Lecture 15)", Math4302_L16: "Modern Algebra (Lecture 16)", Math4302_L17: "Modern Algebra (Lecture 17)", + Math4302_L18: "Modern Algebra (Lecture 18)", }