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圆盘附加到 8 字形环上来获得环面

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hbghlyj posted 2023-7-11 22:06 |Read mode
Exercise 1. A figure 8 loop consists of two circles touching at a point. Show that a torus can be obtained by attaching a disc onto a figure 8 loop.
8 字形环由两个相交的圆组成。证明可以通过将圆盘附加到 8 字形环上来获得环面。
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original poster hbghlyj posted 2023-7-11 22:10
Last edited by hbghlyj 2023-7-11 22:224.01 CW complexes

(3.15) The 2-torus is built by attaching a square to a figure 8. Since the square is topologically a disc, this is a 2-cell attachment. The boundary of the square (disc) is attached in a more interesting way than the previous examples: its boundary runs along the two loops, $a,b$, of the figure 8 in the order $b^{−1}a^{−1}ba$.
Screenshot 2023-07-11 221524.png
A torus is a 2-cell attached to a circle?

Geometric picture of attaching a 2-cell to the 2-torus


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