From 83ada2df2ac8053f32667bcc53304d9ab634b6ef Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Wed, 4 Feb 2026 13:51:19 -0600 Subject: [PATCH] updates --- content/Math4202/Math4202_L10.md | 100 +++++++++++++++++++++++ content/Math4202/_meta.js | 1 + content/Math4302/Math4302_L10.md | 131 +++++++++++++++++++++++++++++++ content/Math4302/_meta.js | 2 + 4 files changed, 234 insertions(+) create mode 100644 content/Math4202/Math4202_L10.md create mode 100644 content/Math4302/Math4302_L10.md diff --git a/content/Math4202/Math4202_L10.md b/content/Math4202/Math4202_L10.md new file mode 100644 index 0000000..ff5d9ae --- /dev/null +++ b/content/Math4202/Math4202_L10.md @@ -0,0 +1,100 @@ +# 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 $\bar{f}(t)=f(1-t)$. + $$ + f*\bar{f}=e_{x_0},\quad \bar{f}*f=e_{x_1} + $$ + $$ + [f]*[\bar{f}]=[e_{x_0}],\quad [\bar{f}]*[f]=[e_{x_1}] + $$ + +
+Proof + +(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. + +$f_t=f(ts)$ $s\in[0,\frac{1}{2}]$. + +$\bar{f}_t=\bar{f}(1-ts)$ $s\in[\frac{1}{2},1]$. + +
+ +> [!CAUTION] +> +> Homeomorphism does not implies homotopy automatically. + +#### Definition for the fundamental group + +The fundamental group of $X$ at $x$ is defined to be + +$$ +(\Pi_1(X,x),*) +$$ \ No newline at end of file diff --git a/content/Math4202/_meta.js b/content/Math4202/_meta.js index f126ca7..383919e 100644 --- a/content/Math4202/_meta.js +++ b/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)", } diff --git a/content/Math4302/Math4302_L10.md b/content/Math4302/Math4302_L10.md new file mode 100644 index 0000000..75385e3 --- /dev/null +++ b/content/Math4302/Math4302_L10.md @@ -0,0 +1,131 @@ +# Math4302 Modern Algebra (Lecture 9) + +## Groups + +### Group homomorphism + +Recall the kernel of a group homomorphism is the set + +$$ +\operatorname{ker}(\phi)=\{a\in G|\phi(a)=e'\} +$$ + +
+Example + +Let $\phi:(\mathbb{Z},+)\to (\mathbb{Z}_n,+)$ where $\phi(k)=k\mod n$. + +The kernel of $\phi$ is the set of all multiples of $n$. + +
+ +#### Theorem for one-to-one group homomorphism + +$\phi:G\to G'$ is one-to-one if and only if $\operatorname{ker}(\phi)=\{e\}$ + +If $\phi$ is one-to-one, then $\phi(G)\leq G'$, $G$ is isomorphic ot $\phi(G)$ (onto automatically). + +If $A$ is a set, then a permutation of $A$ is a bijection $f:A\to A$. + +#### Cayley's Theorem + +Every group $G$ is isomorphic to a subgroup of $S_A$ for some $A$ (and if $G$ is finite then $A$ can be taken to be finite.) + +
+Example + +$D_n\leq S_n$, so $A=\{1,2,\cdots,n\}$ + +--- + +$\mathbb{Z}_n\leq S_n$, (use the set of rotations) so $A=\{1,2,\cdots,n\}$ $\phi(i)=\rho^i$ where $i\in \mathbb{Z}_n$ and $\rho\in D_n$ + +--- + +$GL(2,\mathbb{R})$. Set $A=\mathbb{R}^2$, for every $A\in GL(2,\mathbb{R})$, let $\phi(A)$ be the permutation of $\mathbb{R}^2$ induced by $A$, so $\phi(A)=f_A:\mathbb{R}^2\to \mathbb{R}^2$, $f_A(\begin{pmatrix}x\\y\end{pmatrix})=A\begin{pmatrix}x\\y\end{pmatrix}$ + +We want to show that this is a group homomorphism. + +- $\phi(AB)=\phi(A)\phi(B)$ (it is a homomorphism) + +$$ +\begin{aligned} + f_{AB}(\begin{pmatrix}x\\y\end{pmatrix})&=AB\begin{pmatrix}x\\y\end{pmatrix}\\ + &=f_A(B\begin{pmatrix}x\\y\end{pmatrix})\\ + &=f_A(f_B(\begin{pmatrix}x\\y\end{pmatrix}))\\ + &=(f_A\circ f_B)(\begin{pmatrix}x\\y\end{pmatrix})\\ +\end{aligned} +$$ + +- Then we need to show that $\phi$ is one-to-one. + +It is sufficient to show that $\operatorname{ker}(\phi)=\{e\}$. + +Solve $f_A(\begin{pmatrix}x\\y\end{pmatrix})=\begin{pmatrix}x\\y\end{pmatrix}$, the only choice for $A$ is the identity matrix. + +Therefore $\operatorname{ker}(\phi)=\{e\}$. + +
+ +
+Proof for Cayley's Theorem + +Let $A=G$, for every $g\in G$, define $\lambda_g:G\to G$ by $\lambda_g(x)=gx$. + +Then $\lambda_g$ is a **permutation** of $G$. (not homomorphism) + +- $\lambda_g$ is one-to-one by cancellation on the left. +- $\lambda_g$ is onto since $\lambda_g(g^{-1}y)=y$ for every $y\in G$. + +We claim $\phi: G\to S_G$ define by $\phi(g)=\lambda_g$ is a group homomorphism that is one-to-one. + +First we show that $\phi$ is homomorphism. + +$\forall x\in G$ + +$$ +\begin{aligned} + \phi(g_1)\phi(g_2)&=\lambda_{g_1}(\lambda_{g_2}(x))\\ + &=\lambda_{g_1g_2}(x)\\ + &=\phi(g_1g_2)x\\ +\end{aligned} +$$ + +This is one to one since if $\phi(g_1)=\phi(g_2)$, then $\lambda_{g_1}=\lambda_{g_2}\forall x$, therefore $g_1=g_2$. + +
+ +### Odd and even permutations + +#### Definition of transposition + +A $\sigma\in S_n$ is a transposition is a two cycle $\sigma=(i j)$ + +Fact: Every permutation in $S_n$ can be written as a product of transpositions. (may not be disjoint transpositions) + +
+Example of a product of transpositions + +Consider $(1234)=(14)(13)(12)$. + +In general, $(i_1,i_2,\cdots,i_m)=(i_1i_m)(i_2i_{m-1})(i_3i_{m-2})\cdots(i_1i_2)$ + +This is not the unique way. + +$$ +(12)(34)=(42)(34)(23)(12) +$$ + +
+ +But the parity of the number of transpositions is unique. + +#### Theorem for parity of transpositions + +If $\sigma\in S_n$ is written as a product of transposition, then the number of transpositions is either always odd or even. + +#### Definition of odd and even permutations + +$\sigma$ is an even permutation if the number of transpositions is even. + +$\sigma$ is an odd permutation if the number of transpositions is odd. \ No newline at end of file diff --git a/content/Math4302/_meta.js b/content/Math4302/_meta.js index a02dddb..083e8f1 100644 --- a/content/Math4302/_meta.js +++ b/content/Math4302/_meta.js @@ -11,4 +11,6 @@ export default { Math4302_L6: "Modern Algebra (Lecture 6)", Math4302_L7: "Modern Algebra (Lecture 7)", Math4302_L8: "Modern Algebra (Lecture 8)", + Math4302_L9: "Modern Algebra (Lecture 9)", + Math4302_L10: "Modern Algebra (Lecture 10)", }