# 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 $\varphi_\alpha^1$ is a continuous map that maps the boundary of $e_\alpha^1$ to $X_0$, and $e_\alpha^1$ is a $1$-cell (interval).

$$
X_2=\{(e_\alpha^2,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^2\to X_1\}=(\sqcup_{\alpha\in A}e_\alpha^2)\sqcup X_1
$$

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: every two distinct points of $X$ have disjoint neighborhoods
2. Second countable: With a countable basis
3. Local euclidean: Each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$.

<details>
<summary>Example of space that is not a manifold but satisfies part of the definition</summary>

Non-hausdorff:

Consider the set with two origin $\mathbb{R}\setminus\{0\}$. with $\{p,q\}$, and the topology defined over all the open intervals that don't contain the origin, with set of the form $(-a,0)\cup \{p\}\cup (0,a)$ for $a\in \mathbb{R}$ and $(-a,0)\cup \{q\}\cup (0,a)$.

---

Non-second-countable:

Consider the long line $\mathbb{R}\times [0,1)$

---

Non-local-euclidean:

Any 1-dimensional CW complex (graph) that has a vertex with 3 or more edges connected to it will be Hausdorff and second-countable, but not locally Euclidean at those vertices.

</details>

#### 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 $\{U_i\}_{i=1}^n$.

#### Definition of paracompact space

Locally finite: $\forall x\in X$, $\exists$ open $x\in U$ such that $U$ only intersects finitely many open sets in $\mathcal{B}$.

A space $X$ is paracompact if every open cover $A$ of $X$ has a **locally finite** refinement $\mathcal{B}$ of $A$ that covers $X$.

### 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.

#### 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 $\bar{f}(t)=f(1-t)$.
    $$
    f*\bar{f}=e_{x_0},\quad \bar{f}*f=e_{x_1}
    $$
    $$
    [f]*[\bar{f}]=[e_{x_0}],\quad [\bar{f}]*[f]=[e_{x_1}]
    $$


### Covering space

#### Definition of partition into slice

Let $p:E\to B$ be a continuous surjective map. The open set $U\subseteq B$ is said to be evenly covered by $p$ if it's inverse image $p^{-1}(U)$ can be written as the union of **disjoint open sets** $V_\alpha$ in $E$. Such that for each $\alpha$, the restriction of $p$ to $V_\alpha$ is a homeomorphism of $V_\alpha$ onto $U$.

The collection of $\{V_\alpha\}$ is called a **partition** $p^{-1}(U)$ into slice.

#### 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 $\mathbb{Z}$.

#### Theorem for induced homotopy for fundamental groups

Suppose $f,g$ are two paths in $B$, and suppose $f$ and $g$ are path homotopy ($f(0)=g(0)=b_0$, and $f(1)=g(1)=b_1$, $b_0,b_1\in B$), then $\hat{f}:\pi_1(B,b_0)\to \pi_1(B,b_1)$ and $\hat{g}:\pi_1(B,b_0)\to \pi_1(B,b_1)$ are path homotopic.