diff --git a/content/Math4202/Exam_reviews/Math4202_E1.md b/content/Math4202/Exam_reviews/Math4202_E1.md new file mode 100644 index 0000000..3eb9340 --- /dev/null +++ b/content/Math4202/Exam_reviews/Math4202_E1.md @@ -0,0 +1,152 @@ +# 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. + +$$ +X_n=\{(e_\alpha^n,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^n\to X_{n-1}\}=(\sqcup_{\alpha\in A}e_\alpha^n)\sqcup X_{n-1} +$$ + +
+Example of CW complex construction + +$X_0=a$ + +$X_1=$ 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 $a$ +
+ +## 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 $\bar{f}(t)=f(1-t)$. + $$ + f*\bar{f}=e_{x_0},\quad \bar{f}*f=e_{x_1} + $$ + $$ + [f]*[\bar{f}]=[e_{x_0}],\quad [\bar{f}]*[f]=[e_{x_1}] + $$ \ No newline at end of file diff --git a/content/Math4202/Exam_reviews/Math4202_P1.md b/content/Math4202/Exam_reviews/Math4202_P1.md new file mode 100644 index 0000000..a07964d --- /dev/null +++ b/content/Math4202/Exam_reviews/Math4202_P1.md @@ -0,0 +1,26 @@ +# 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). \ No newline at end of file diff --git a/content/Math4202/Math4202_L10.md b/content/Math4202/Math4202_L10.md index ff5d9ae..49fa9c8 100644 --- a/content/Math4202/Math4202_L10.md +++ b/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) diff --git a/content/Math4202/Math4202_L11.md b/content/Math4202/Math4202_L11.md index 9e6fe2d..de0b896 100644 --- a/content/Math4202/Math4202_L11.md +++ b/content/Math4202/Math4202_L11.md @@ -81,7 +81,7 @@ $\Pi_1(X,1)=\mathbb{Z}$ (related to winding numbers, prove next week). -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 $\hat{\alpha}$ diff --git a/content/Math4202/_meta.js b/content/Math4202/_meta.js index a80187b..f261ea4 100644 --- a/content/Math4202/_meta.js +++ b/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)",