From b4ff21e4008aeda6ccc7fa42d55f633951c8e9f8 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Sun, 1 Mar 2026 21:34:40 -0600 Subject: [PATCH] udpates --- content/Math4202/Exam_reviews/Math4202_E1.md | 40 ++++++++++++++++- content/Math4202/Exam_reviews/Math4202_P1.md | 45 +++++++++++++++++--- 2 files changed, 79 insertions(+), 6 deletions(-) diff --git a/content/Math4202/Exam_reviews/Math4202_E1.md b/content/Math4202/Exam_reviews/Math4202_E1.md index 3eb9340..3c316df 100644 --- a/content/Math4202/Exam_reviews/Math4202_E1.md +++ b/content/Math4202/Exam_reviews/Math4202_E1.md @@ -149,4 +149,42 @@ $$ $$ $$ [f]*[\bar{f}]=[e_{x_0}],\quad [\bar{f}]*[f]=[e_{x_1}] - $$ \ No newline at end of file + $$ + + +### 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}$. diff --git a/content/Math4202/Exam_reviews/Math4202_P1.md b/content/Math4202/Exam_reviews/Math4202_P1.md index a07964d..2bdabdc 100644 --- a/content/Math4202/Exam_reviews/Math4202_P1.md +++ b/content/Math4202/Exam_reviews/Math4202_P1.md @@ -6,21 +6,56 @@ 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}$. + +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$. -- (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$ by $x\to x/\|x\|$. + +## 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). \ No newline at end of file + 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. \ No newline at end of file