# 拓扑学I|Topology I代写 MATH 671

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So map each simplex in the first barycentric subdivision of $K$,
$$\sigma_{1}<\sigma_{2}<\cdots<\sigma_{n}, \quad \sigma_{i} \in K$$
to the simplex in nerve $C(U)$,
$$U_{\sigma_{n}} \subseteq U_{\sigma_{n-1}} \subseteq \cdots \subseteq U_{\sigma_{1}}$$

This gives a simplicial map
$K^{\prime} \rightarrow \vec{g}$ nerve $C(U), K^{\prime}=1$ st barycentric subdivision $.$
Now consider the compositions
\begin{aligned} &K^{\prime} \underset{g}{\rightarrow} \text { nerve } C(U) \underset{f}{\rightarrow} K \ &\text { nerve } C(U) \stackrel{f}{\rightarrow} K=K^{\prime} \stackrel{g}{\rightarrow} \text { nerve } C(U) \end{aligned}
One can check for the first composition that a simplex of $K^{\prime}$
$$\sigma_{1}<\cdots<\sigma_{n}$$

## MATH 671 COURSE NOTES ：

(the two sphere, $S^{2}$ ) Let $p$ and $\sigma$ denote two distinct points of $S^{2}$. Consider the category over $S^{2}$ determined by the maps
$$\begin{array}{cc} \text { object } & \text { name } \ S^{2}-p \stackrel{\subseteq}{\longrightarrow} S^{2} & e \ S^{2}-q \subseteq S^{2} & e^{\prime} \ \text { (universal cover } \left.S^{2}-p-q\right) \rightarrow S^{2} & \mathbb{Z} . \end{array}$$
The category has three objects and might be denoted