63 lines
2.4 KiB
Markdown
63 lines
2.4 KiB
Markdown
# Math4202 Topology II (Lecture 7)
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## Algebraic Topology
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Classify 2-dimensional topological manifolds (connected) up to homeomorphism/homotopy equivalence.
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Use fundamental groups.
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We want to show that:
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1. The fundamental group is invariant under the equivalence relation.
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2. develop some methods to compute the groups.
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3. 2-dimensional topological spaces with the same fundamental group are equivalent (homeomorphism).
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### Homotopy of paths
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#### Definition of path
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If $f$ and $f'$ are two continuous maps from $X$ to $Y$, where $X$ and $Y$ are topological spaces. Then we say that $f$ is homotopic to $f'$ if there exists a continuous map $F:X\times [0,1]\to Y$ such that $F(x,0)=f(x)$ and $F(x,1)=f'(x)$ for all $x\in X$.
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The map $F$ is called a homotopy between $f$ and $f'$.
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We use $f\simeq f'$ to mean that $f$ is homotopic to $f'$.
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#### Definition of homotopic equivalence map
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Let $f:X\to Y$ and $g:Y\to X$ be two continuous maps. If $f\circ g:Y\to Y$ and $g\circ f:X\to X$ are homotopic to the identity maps $\operatorname{id}_Y$ and $\operatorname{id}_X$, then $f$ and $g$ are homotopic equivalence maps. And the two spaces $X$ and $Y$ are homotopy equivalent.
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> [!NOTE]
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>
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> This condition is weaker than homeomorphism. (In homeomorphism, let $g=f^{-1}$, we require $g\circ f=\operatorname{id}_X$ and $f\circ g=\operatorname{id}_Y$.)
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<details>
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<summary>Example of homotopy equivalence maps</summary>
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Let $X=\{a\}$ and $Y=[0,1]$ with standard topology.
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Consider $f:X\to Y$ by $f(a)=0$ and $g:Y\to X$ by $g(y)=a$, where $y\in [0,1]$.
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$g\circ f=\operatorname{id}_X$ and $f\circ g=[0,1]\mapsto 0$.
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$g\circ f\simeq \operatorname{id}_X$
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and $f\circ g\simeq \operatorname{id}_Y$.
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Consider $F:X\times [0,1]\to Y$ by $F(a,0)=0$ and $F(a,t)=(1-t)y$. $F$ is continuous and homotopy between $f\circ g$ and $\operatorname{id}_Y$.
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This gives example of homotopy but not homeomorphism.
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</details>
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#### Definition of null homology
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If $f:X\to Y$ is homotopy to a constant map. $f$ is called null homotopy.
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#### Definition of path homotopy
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Let $f,f':I\to X$ be a continuous maps from an interval $I=[0,1]$ to a topological space $X$.
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Two pathes $f$ and $f'$ are path homotopic if
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- 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 $i\in I$.
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- $F(s,0)=f(0)$ and $F(s,1)=f(1)$, $\forall s\in I$. |