Merge branch 'main' of https://git.trance-0.com/Trance-0/NoteNextra-origin

content/Math4202/Exam_reviews/Math4202_E1.md

@@ -34,7 +34,7 @@ $$
 X_1=\{(e_\alpha^1,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^1\to X_0\}
 $$
 
-where $\varphi$ is a continuous map, and $e_\alpha^1$ is a $1$-cell (interval).
+where $\varphi_\alpha^1$ is a continuous map that maps the boundary of $e_\alpha^1$ to $X_0$, and $e_\alpha^1$ is a $1$-cell (interval).
 
 $$
 X_2=\{(e_\alpha^2,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^2\to X_1\}=(\sqcup_{\alpha\in A}e_\alpha^2)\sqcup X_1
@@ -74,10 +74,9 @@ $X_2=$ ballon shape with boundary of circle collapsing at $a$
 
 An $m$-dimensional **manifold** is a topological space  $X$ that is
 
-1. Hausdorff
+1. Hausdorff: every two distinct points of $X$ have disjoint neighborhoods
-2. With a countable basis
+2. Second countable: With a countable basis
-3. Each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$. (local euclidean)
+3. Local euclidean: Each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$.
-
 
 #### Whitney's Embedding Theorem
 
@@ -100,6 +99,18 @@ Then there exists a partition of unity dominated by $\{U_i\}_{i=1}^n$.
 
 ### Homotopy
 
+#### Definition of homotopy equivalent spaces
+
+Let $f:X\to Y$ and $g:X\to Y$ be tow continuous maps from a topological space $X$ to a topological space $Y$.
+
+$f\circ g:Y\to Y$ should be homotopy to $Id_Y$ and $g\circ f:X\to X$ should be homotopy to $Id_X$.
+
+#### Definition of homotopy
+
+Let $f:X\to Y$ and $g:X\to Y$ be tow continuous maps from a topological space $X$ to a topological space $Y$.
+
+If there exists a continuous map $F:X\times [0,1]\to Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x\in X$, then $f$ and $g$ are homotopy equivalent.
+
 #### Definition of null homology
 
 If $f:X\to Y$ is homotopy to a constant map. $f$ is called null homotopy.
@@ -149,4 +160,42 @@ $$
     $$
     $$
     [f]*[\bar{f}]=[e_{x_0}],\quad [\bar{f}]*[f]=[e_{x_1}]
-    $$
+    $$
+
+
+### Covering space
+
+#### Definition of covering space
+
+Let $p:E\to B$ be a continuous surjective map.
+
+If every point $b$ of $B$ has a neighborhood **evenly covered** by $p$, which means $p^{-1}(U)$ is a union of disjoint open sets, then $p$ is called a covering map and $E$ is called a covering space.
+
+#### Theorem exponential map gives covering map
+
+The map $p:\mathbb{R}\to S^1$ defined by $x\mapsto e^{2\pi ix}$ or $(\cos(2\pi x),\sin(2\pi x))$ is a covering map.
+
+#### Definition of local homeomorphism
+
+A continuous map $p:E\to B$ is called a local homeomorphism if for **every $e\in E$** (note that for covering map, we choose $b\in B$), there exists a neighborhood $U$ of $b$ such that $p|_U:U\to p(U)$ is a homeomorphism on to an open subset $p(U)$ of $B$.
+
+Obviously, every open map induce a local homeomorphism. (choose the open disk around $p(e)$)
+
+#### Theorem for subset covering map
+
+Let $p: E\to B$ be a covering map. If $B_0$ is a subset of $B$, the map $p|_{p^{-1}(B_0)}: p^{-1}(B_0)\to B_0$ is a covering map.
+
+#### Theorem for product of covering map
+
+If $p:E\to B$ and $p':E'\to B'$ are covering maps, then $p\times p':E\times E'\to B\times B'$ is a covering map.
+
+
+### Fundamental group of the circle
+
+Recall from previous lecture, we have unique lift for covering map.
+
+#### Lemma for unique lifting for covering map
+
+Let $p: E\to B$ be a covering map, and $e_0\in E$ and $p(e_0)=b_0$. Any path $f:I\to B$ beginning at $b_0$, has a unique lifting to a path starting at $e_0$.
+
+Back to the circle example, it means that there exists a unique correspondence between a loop starting at $(1,0)$ in $S^1$ and a path in $\mathbb{R}$ starting at $0$, ending in $\mathbb{Z}$.

content/Math4202/Exam_reviews/Math4202_P1.md

@@ -6,21 +6,89 @@ In the following, please provide complete proof of the statements and the answer
 
 - (2 points) State the definition of a topological manifold.
 
+A topological manifold is a topological space that satisfies the following:
 
+1. It is Hausdorff
+2. It has a countable basis
+3. Each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$.
+
+- (2 points) Prove that real projective space $\mathbb{R}P^2$ is a manifold.
+
+Let $\mathbb{R}P^2=\mathbb{R}^3/\sim$ where $(x,y,z)\sim(x',y',z')$ if $\lambda(x,y,z)=(x',y',z')$ for some $\lambda\in \mathbb{R},\lambda\neq 0$.
+
+1. It is Hausdorff since $\mathbb{R}^3$ is Hausdorff, subspace of Hausdorff space is Hausdorff.
+2. It has a countable basis since $\mathbb{R}^3$ has a countable basis, subspace of countable basis has countable basis.
+3. Each point of $x$ of $RP^2$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^3$. Let $p$ be an arbitrary point in $RP^2$, Consider the projection on to the tangent plane of $p$ defined as $\mathbb{R}P^2\to \mathbb{R}^2$.
+
+<details>
+<summary>Solution on class</summary>
+
+Consider $\mathbb{R} P^n$ be the lines in $\mathbb{R}^{n+1}$ through the origin.
+
+$$
+\mathbb{R}P^n=\{v\neq 0|v\in \mathbb{R}^{n+1}\}/\sim
+$$
+
+where $a\sim b$ if there exists $\lambda\in \mathbb{R},\lambda\neq 0$ such that $\lambda a=b$.
+
+$$
+S^n=\{v\in \mathbb{R}^{n+1}|||v||=1\}
+$$
+
+First we test the local euclidean structure.
+
+Consider the hemisphere cap $U_{1,+}=\{(x_1,\dots,x_{n+1})|x_1>0\}$, note that this cap induce a quotient mapping to some open set of $\mathbb{R}P^n$
+
+Note that the cap $U_{1,+}$ is local euclidean by the bijective projection map to $\mathbb{R}^n$ $(x_1,\dots,x_{n+1})\mapsto(x_2,\dots,x_{n+1})$.
+
+And with $U_{1,-},U_{2,+},U_{2,-},\dots,U_{n,+},U_{n,-}$ we can construct a open cover of $\mathbb{R}P^n$. Since for any of the point in $\mathbb{R} P^n$ we can have some non-zero coordinates that projects to $S^n$ and we can build such cap.
+
+Second we show the second countability.
+
+Take the cap with rational coordinates, and this creates a countable basis.
+
+Third we prove the Hausdorff property.
+
+Consider $x=(x_1,\dots,x_{n+1})\in \mathbb{R}P^n$, $y=(y_1,\dots,y_{n+1})\in \mathbb{R}P^n$.
+
+</details>
 
-- (2 points) Prove that real projective space $RP^2$ is a manifold.
 - (2 points) Find a 2-1 covering space of $RP^2$.
 
-Problem 2
+Take $\mathbb{R}P^2\to S^2$ with quotient topology where $v\sim -v$.
+
+## Problem 2
+
 - (2 points) State the definition of a CW complex.
+
+Let $X_0$ be arbitrary set of points, and $X_n$ be a CW complex defined by $X_n=\{(e_\alpha^n,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^n\to X_{n-1}\}=(\sqcup_{\alpha\in A}e_\alpha^n)\sqcup X_{n-1}$
+
 - (4 points) Describe a CW complex homeomorphic to the 2-torus.
 
-Problem 3
+Take two points $a,b$, connect $a,b$ with two lines, and add $a$ with a circle connecting to itself, $b$ with a circle connecting to itself. Then wrap a 2-cell on that.
+
+## Problem 3
+
 - (2 points) State the definition of the fundamental group of a topological space $X$ relative to $x_0 \in X$.
+
+The fundamental group of $X$ relative to $x_0$ is the group of all continuous paths from $x_0$ to $x_0$ under path homotopy equivalence.
+
 - (4 points) Compute the fundamental group of $R^n$ relative to the origin.
 
-Problem 4
+The fundamental group of $R^n$ relative to the origin is the trivial group.
+
+## Problem 4
+
 - (2 points) Give a pair of spaces that are homotopic equivalent, but not homeomorphic.
+
+$\mathbb{R}$ and one point set is homotopic equivalent, (using contraction), but not homeomorphic.
+
 - (4 points) Let $A$ be a subspace of $R^n$, and $h : (A, a_0) \to (Y, y_0)$. Show that if $h$ is extendable to a continuous map of $R^n$ into $Y$, then
   $$h_* : \pi_1(A, a_0) \to \pi_1(Y, y_0)$$
-  is the trivial homomorphism (the homomorphism that maps everything to the identity element).
+  is the trivial homomorphism (the homomorphism that maps everything to the identity element).
+
+Since $h$ is extendable to a continuous map of $\R^n$ into $Y$, consider the continuous function $H:(\R^n, x_0)\to (Y,y_0)$, with $H|_{A}(f)=h(f)$.
+
+Note that the inclusion map $i:(A,x_0)\to (\R^n,x_0)$ induces $i_*$ gives a homomorphism, therefore $H\circ i=h$ is a homomorphism. Then $h_*=H_*\circ i_*$. where $\pi_1(\R^n,x_0)$ is trivial since $\R^n$ is contractible.
+
+Thus $H_*$ is the trivial homomorphism. Therefore $h_*$ is the trivial homomorphism.

content/Math4302/Math4302_L20.md

@@ -0,0 +1,148 @@
+# Math4302 Modern Algebra (Lecture 20)
+
+## Groups
+
+### Commutator of a group
+
+Let $G$ be a group and $a,b\in G$, $[a,b]=aba^{-1}b^{-1}$.
+
+Let $G'\leq G$ be the subgroup of $G$ generated by all commutators of $G$, $G'=\{[a_1,b_1][a_2,b_2]\ldots[a_n,b_n]\mid a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n\in G\}$.
+
+Last time we shed that $G'$ is a normal subgroup of $G$ and $G/G'$ is abelian.
+
+#### Proposition for commutator subgroup
+
+If $N\trianglelefteq G$ is a normal subgroup of $G$ and $G/N$ is abelian, then $G'\leq N$.
+
+<details>
+<summary>Proof</summary>
+
+We have $aNbN=bNaN$ for all $a,b\in G$.
+
+so $abN=baN$, $a^{-1}b^{-1}abN=N$, so $a^{-1}b^{-1}\in N$, so for every $a,b\i G$, since $a^{-1},b^{-1}\in N$, $(a^{-1})^{-1}(b^{-1})^{-1}a^{-1}b^{-1}\in N$, so $[a,b]\in N$.
+
+So $G'\leq N$.
+
+</details>
+
+<details>
+<summary>Example</summary>
+
+Consider $G=S_3$. find $G'$.
+
+_the trivial way is to enumerate all the commutators, but we can do better than that applying the proposition above to check if any normal group has an abelian factor group to narrow down the search_
+
+Let $N=\{e,\rho,\rho^2\}$, $\rho=(1,2,3)$, then $|N|=|G|/2$, so $N$ is normal and $|G/N|=2 so $G/N\simeq \mathbb{Z}_2$ si abelian, so $G'\subseteq N$.
+
+Now let $\rho=(1,2,3),\sigma=(1,2)$, $[\rho,\sigma]=(1,2,3)(1,2)(1,3,2)(1,2)=(1,3,2)$
+
+So $\rho^2=(1,3,2)$ is in $G'$ and $\rho=(1,3,2)^{-1}\in G$, therefore $N\subseteq G'$.
+
+So $G'=N$.
+
+> Few additional exercises to for $n\geq 5$, we have $G'=A_n$. (relates to simple subgroup properties.) You may check it out.
+
+</details>
+
+### Group acting on a set
+
+#### Definition for group acting on a set
+
+Let $G$ be a group, $X$ be a set, $X$ is a $G$-set or $G$ acts on $X$ if there is a map
+
+$$
+G\times X\to X
+$$
+$$
+(g,x)\mapsto g\cdot x\, (\text{ or simply }g(x))
+$$
+
+such that
+
+1. $e\cdot x=x,\forall x\in X$
+2. $g_2\cdot(g_1\cdot x)=(g_2 g_1)\cdot x$
+
+> There is always a trivial action defined on $X$ by $g\cdot x=x$ satisfying the two properties.
+
+<details>
+<summary>Example</summary>
+
+Let $G$ be a group,
+
+$G$ acts on $G$ by $g\cdot x\coloneqq g h$, $g,x\in G$
+
+---
+
+$G$ acts on $G$ via conjugation, $g\cdot x\coloneqq g x g^{-1}$, $g,x\in G$
+
+Let's check the two properties are satisfied.
+
+$e\cdot x=exe^{-1}=x$
+
+$$
+\begin{aligned}
+g_2\cdot (g_1\cdot x)&= g_2\cdot (g_1xg_1^{-1})\\
+&= g_2g_1xg_1^{-1} g_2^{-1}\\
+&= g_2g_1xg_1^{-1} g_2^{-1}\\
+&= (g_2 g_1)(x)(g_1^{-1}g_2^{-1})\\
+&= (g_2 g_1)(x)
+\end{aligned}
+$$
+
+---
+
+Take $S_n$ acts on $\{1,2,\ldots,n\}$ via $\sigma\cdot x\coloneqq \sigma(x)$.
+
+---
+
+$GL(n,\mathbb{R})$ (general linear group) acts on $\mathbb{R}^n$ by $A\cdot x\coloneqq A x$, $A\in GL(n,\mathbb{R}), x\in \mathbb{R}^n$
+
+</details>
+
+#### Group action is a homomorphism
+
+Let $X$ be a $G$-set, $g\in G$, then the function
+
+$$
+\sigma_g:X\to X,x\mapsto g\cdot x
+$$
+
+is a bijection, and the function $\phi:G\to S_X, g\mapsto \sigma_g$ is a group homomorphism.
+
+<details>
+
+<summary>Proof</summary>
+
+$\sigma_g$ is onto: If $y\in X$, let $x=g^{-1}y$, then $\sigma_g(g^{-1}\cdot y)=g\cdot (g^{-1}\cdot y)=(gg^{-1})\cdot y=e\cdot y=y$.
+
+$\sigma_g$ is one-to-one: If $\sigma_g(x_1)=\sigma_g(x_2)$, then $g\cdot x_1=g\cdot x_2$.
+
+So $g^{-1}\cdot (g\cdot x_1)=g^{-1}\cdot (g\cdot x_2)=x_1=x_2$.
+
+Then we need to show that $\phi$ is a homomorphism.
+
+$$
+\phi(g_1g_2)=\phi(g_1)\cdot \phi(g_2)
+$$
+
+Note that $\phi(g_1g_2)=\sigma_{g_1g_2}$, $\phi(g_1)=\sigma_{g_1}$, $\phi(g_2)=\sigma_{g_2}$.
+
+For every $x\in X$, $\sigma_{g_1g_2}(x)=(g_1g_2)\cdot x$, $\sigma_{g_1}(x)=g_1\cdot x$, $\sigma_{g_2}(x)=g_2\cdot x$. By the second property of $G$-sets, we have $\sigma_{g_1}\cdot \sigma_{g_2}=g_1\circ(g_2\circ x)=(g_1g_2)\circ x=\sigma_{g_1g_2}\circ x$.
+
+</details>
+
+> [!NOTE]
+>
+> $\phi$ as above is general not injective and not surjective.
+>
+> If $G$ acts trivially on $x$ ($g\cdot x=x,\forall g\in G$), then $\phi(g)$ is the identity function for all $g\in G$.
+
+Define a relation on $X$ by $x\sim y\iff y=g\cdot x$ for some $g\in G$.
+
+This equivalence relation is well-defined.
+
+- Reflexive: $x\sim x$, take $e\cdot x$
+- Symmetric: $x\sim y\implies y\sim x$ ($g^{-1}\in G$, $g^{-1}\cdot (g\cdot x)=g^{-1}\cdot y$)
+- Transitive: $x\sim y, y\sim z\implies x\sim z$ take $x=g_1\cdot y=g_1\cdot (g_2\cdot z)=g_1g_2\cdot z$ and $g_1g_2\in G$.
+
+This gives a orbit for $x\in X$!

content/Math4302/_meta.js

@@ -22,4 +22,5 @@ export default {
     Math4302_L17: "Modern Algebra (Lecture 17)",
     Math4302_L18: "Modern Algebra (Lecture 18)",
     Math4302_L19: "Modern Algebra (Lecture 19)",
+    Math4302_L20: "Modern Algebra (Lecture 20)",
 }