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content/Math4202/Exam_reviews/Math4202_E1.md

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+# Math4202 Topology II Exam 1 Review
+
+> [!NOTE]
+>
+> This is a review for definitions we covered in the classes. It may serve as a cheat sheet for the exam if you are allowed to use it.
+
+## Few important definitions
+
+### Quotient spaces
+
+Let $X$ be a topological space and $f:X\to Y$ is a
+
+1. continuous
+2. surjective map.
+3. With the property that $U\subset Y$ is open if and only if $f^{-1}(U)$ is open in $X$.
+
+Then we say $f$ is a quotient map and $Y$ is a quotient space.
+
+#### Theorem of quotient space
+
+Let $p:X\to Y$ be a quotient map, let $Z$ be a space and $g:X\to Z$ be a map that is constant on each set $p^{-1}(y)$ for each $y\in Y$.
+
+Then $g$ induces a map $f: X\to Z$ such that $f\circ p=g$.
+
+The map $f$ is continuous if and only if $g$ is continuous; $f$ is a quotient map if and only if $g$ is a quotient map.
+
+### CW complex
+
+Let $X_0$ be arbitrary set of points.
+
+Then we can create $X_1$ by
+
+$$
+X_1=\{(e_\alpha^1,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^1\to X_0\}
+$$
+
+where $\varphi$ is a continuous map, and $e_\alpha^1$ is a $1$-cell (interval).
+
+$$
+X_2=\{(e_\alpha^2,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^2\to X_1\}=(\sqcup_{\alpha\in A}e_\alpha^2)\sqcup X_1
+$$
+
+and $e_\alpha^2$ is a $2$-cell (disk). (mapping boundary of disk to arc (like a croissant shape, if you try to preserve the area))
+
+The higher dimensional folding cannot be visualized in 3D space.
+
+$$
+X_n=\{(e_\alpha^n,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^n\to X_{n-1}\}=(\sqcup_{\alpha\in A}e_\alpha^n)\sqcup X_{n-1}
+$$
+
+<details>
+<summary>Example of CW complex construction</summary>
+
+$X_0=a$
+
+$X_1=$ circle, with end point and start point at $a$
+
+$X_2=$ sphere (shell only), with boundary shrinking at the circle create by $X_1$
+
+---
+
+$X_0=a$
+
+$X_1=a$
+
+$X_2=$ ballon shape with boundary of circle collapsing at $a$
+</details>
+
+## Algebraic topology
+
+### Manifold
+
+#### Definition of Manifold
+
+An $m$-dimensional **manifold** is a topological space  $X$ that is
+
+1. Hausdorff
+2. 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)
+
+
+#### Whitney's Embedding Theorem
+
+If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^N$ for some positive integer $N$.
+
+_In general, $X$ is not required to be compact. And $N$ is not too big. For non compact $X$, $N\leq 2m+1$ and for compact $X$, $N\leq 2m$._
+
+#### Definition for partition of unity
+
+Let $\{U_i\}_{i=1}^n$ be a finite open cover of topological space $X$. An indexed family of **continuous** function $\phi_i:X\to[0,1]$ for $i=1,...,n$ is said to be a **partition of unity** dominated by $\{U_i\}_{i=1}^n$ if
+
+1. $\operatorname{supp}(\phi_i)=\overline{\{x\in X: \phi_i(x)\neq 0\}}\subseteq U_i$ (the closure of points where $\phi_i(x)\neq 0$ is in $U_i$) for all $i=1,...,n$
+2. $\sum_{i=1}^n \phi_i(x)=1$ for all $x\in X$ (partition of function to $1$)
+
+#### Existence of finite partition of unity
+
+Let $\{U_i\}_{i=1}^n$ be a finite open cover of a normal space $X$ (Every pair of closed sets in $X$ can be separated by two open sets in $X$).
+
+Then there exists a partition of unity dominated by $\{U_i\}_{i=1}^n$.
+
+### Homotopy
+
+#### Definition of null homology
+
+If $f:X\to Y$ is homotopy to a constant map. $f$ is called null homotopy.
+
+#### Definition of path homotopy
+
+Let $f,f':I\to X$ be a continuous maps from an interval $I=[0,1]$ to a topological space $X$.
+
+Two pathes $f$ and $f'$ are path homotopic if
+
+- there exists a continuous map $F:I\times [0,1]\to X$ such that $F(i,0)=f(i)$ and $F(i,1)=f'(i)$ for all $i\in I$.
+- $F(s,0)=f(0)$ and $F(s,1)=f(1)$, $\forall s\in I$.
+
+#### Lemma: Homotopy defines an equivalence relation
+
+The $\simeq$, $\simeq_p$ are both equivalence relations.
+
+
+#### Definition for product of paths
+
+Given $f$ a path in $X$ from $x_0$ to $x_1$ and $g$ a path in $X$ from $x_1$ to $x_2$.
+
+Define the product $f*g$ of $f$ and $g$ to be the map $h:[0,1]\to X$.
+
+#### Definition for equivalent classes of paths
+
+$\Pi_1(X,x)$ is the equivalent classes of paths starting and ending at $x$.
+
+On $\Pi_1(X,x)$,, we define $\forall [f],[g],[f]*[g]=[f*g]$.
+
+$$
+[f]\coloneqq \{f_i:[0,1]\to X|f_0(0)=f(0),f_i(1)=f(1)\}
+$$
+
+#### 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 $\bar{f}(t)=f(1-t)$.
+    $$
+    f*\bar{f}=e_{x_0},\quad \bar{f}*f=e_{x_1}
+    $$
+    $$
+    [f]*[\bar{f}]=[e_{x_0}],\quad [\bar{f}]*[f]=[e_{x_1}]
+    $$

content/Math4202/Exam_reviews/Math4202_P1.md

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+# Math4202 Topology II Exam 1 Practice
+
+In the following, please provide complete proof of the statements and the answers you give. The total score is 25 points.
+
+## Problem 1
+
+- (2 points) State the definition of a topological manifold.
+
+
+
+- (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
+- (2 points) State the definition of a CW complex.
+- (4 points) Describe a CW complex homeomorphic to the 2-torus.
+
+Problem 3
+- (2 points) State the definition of the fundamental group of a topological space $X$ relative to $x_0 \in X$.
+- (4 points) Compute the fundamental group of $R^n$ relative to the origin.
+
+Problem 4
+- (2 points) Give a pair of spaces that are homotopic equivalent, 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).

content/Math4202/Math4202_L10.md

@@ -4,7 +4,6 @@
 
 ### 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)

content/Math4202/Math4202_L11.md

@@ -81,7 +81,7 @@ $\Pi_1(X,1)=\mathbb{Z}$ (related to winding numbers, prove next week).
 
 </details>
 
-A natural question is, will the fundamental group depends on the basepoint $x$?
+A natural question is, will the fundamental group depends on the base point $x$?
 
 #### Definition for $\hat{\alpha}$
 

content/Math4202/_meta.js

@@ -3,6 +3,7 @@ export default {
     "---":{
         type: 'separator'
     },
+    Exam_reviews: "Exam reviews",
     Math4202_L1: "Topology II (Lecture 1)",
     Math4202_L2: "Topology II (Lecture 2)",
     Math4202_L3: "Topology II (Lecture 3)",