29 lines
1.0 KiB
Markdown
29 lines
1.0 KiB
Markdown
# Math4202 Topology II (Lecture 30)
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## Algebraic Topology
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We skipped a few chapters about Jordan curve theorem, which will be your final project soon. LOL, I will embedded the link once I'm done.
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### Seifert-Van Kampen Theorem
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#### The Seifert-Van Kampen Theorem
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Let $X=U\cup V$ be a union of two open subspaces. Suppose that $U\cap V$, $U,V$ are path connected. Fix $x_0\in U\cap V$.
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Let $H$ be a group (arbitrary). And now we assume $\phi_1,\phi_2$ be a group homomorphism, and $\phi_1:\pi_1(U,x_0)\to H$, and $\phi_2:\pi_1(V,x_0)\to H$.
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Let $i_1,i_2,j_1,j_2,i_{12}$ be group homomorphism induced by the inclusion maps.
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Assume this diagram commutes.
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$$
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\phi_1\circ i_1=\phi_2\circ i_2
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$$
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There is a group homomorphism $\Phi:\pi_1(X,x_0)\to H$ making the diagram commute. $\Phi\circ j_1=\phi_1$ and $\Phi\circ j_2=\phi_2$.
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We may change the base point using conjugations.
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