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_\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}

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

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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: 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.

Example of space that is not a manifold but satisfies part of the definition

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).

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Non-second-countable:

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

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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.

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.