NoteNextra-origin/content/Math4202/Exam_reviews/Math4202_E1.md

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}]