udpates

content/Math4202/Exam_reviews/Math4202_E1.md

@@ -149,4 +149,42 @@ $$
     $$
     $$
     [f]*[\bar{f}]=[e_{x_0}],\quad [\bar{f}]*[f]=[e_{x_1}]
-    $$
+    $$
+
+
+### 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 +6,56 @@ 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).
+  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.