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83 lines
2.3 KiB
Markdown
83 lines
2.3 KiB
Markdown
# Math4202 Topology II (Lecture 3)
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## Reviewing quotient map
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### Quotient map from equivalence relation
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Consider $X,Y$ be two topological space and $A\subset X$, where $f:A\to Y$ is a function.
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Then the disjoint union $X\sqcup Y /_{a\sim f(a)}$ is a quotient space of $X\sqcup Y$ by the equivalence relation $a\sim f(a)$
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Consider $e^n$ be the n dimensional closed ball (n-cells)
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$$
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e^n=\{x\in \mathbb{R}^n:\sum_{i=1}^n x_i^2\leq 1\}
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$$
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and $\partial e^n=A$ be the $n-1$ dimensional sphere.
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#### CW complex
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Let $X_0$ be arbitrary set of points.
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Then we can create $X_1$ by
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$$
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X_1=\{(e_\alpha^1,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^1\to X_0\}
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$$
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where $\varphi$ is a continuous map, and $e_\alpha^1$ is a $1$-cell (interval).
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$$
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X_2=\{(e_\alpha^2,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^2\to X_1\}=(\sqcup_{\alpha\in A}e_\alpha^2)\sqcup X_1
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$$
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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))
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The higher dimensional folding cannot be visualized in 3D space.
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$$
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X_n=\{(e_\alpha^n,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^n\to X_{n-1}\}=(\sqcup_{\alpha\in A}e_\alpha^n)\sqcup X_{n-1}
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$$
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<details>
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<summary>Example of CW complex construction</summary>
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$X_0=a$
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$X_1=$ circle, with end point and start point at $a$
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$X_2=$ sphere (shell only), with boundary shrinking at the circle create by $X_1$
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---
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$X_0=a$
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$X_1=a$
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$X_2=$ ballon shape with boundary of circle collapsing at $a$
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</details>
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#### Theorem of quotient space
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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 $y\in Y$.
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Then $g$ induces a map $f: X\to Z$ such that $f\circ p=g$.
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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.
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## Imbedding of Manifolds
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### Manifold
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#### Definition of Manifold
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An $m$-dimensional **manifold** is a topological space $X$ that is
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1. Hausdorff
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2. With a countable basis such that each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$.
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> [!NOTE]
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>
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> Try to find some example that satisfies some of the properties above but not a manifold.
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