# 拓扑学 Topology MATH346

When $W$ is finite, $L_{T}$ is a simplex for each $T \neq S$ and $L_{S}=\emptyset$. Hence,
$$1-\chi\left(L_{T}\right)= \begin{cases}0 & \text { if } T \neq S, \ 1 & \text { if } T=S .\end{cases}$$
So when $W$ is finite the theorem is the tautology $1 / W(\mathbf{t})=1 / W(\mathbf{t})$.
Suppose $W$ is infinite. We can rewrite Corollary (ii) as
$$\frac{1}{W(t)}=-\varepsilon(S) \sum_{T \subsetneq S} \frac{\varepsilon(T)}{W_{T}(t)}$$
The proof is by induction on $\operatorname{Card}(T)$. For any $T \subset S$, let $\mathcal{S}(T)$ be the set of spherical subsets of $T$ and for any $U \in \mathcal{S}(T)$, let $L_{U}(T)$ be the simplicial complex corresponding to $\mathcal{S}(T), U$. Using (17.11) and the inductive hypothesis, we get
$$\frac{1}{W(\mathbf{t})}=-\varepsilon(S) \sum_{T \subsetneq S} \varepsilon(T) \sum_{U \in \mathcal{S}(T)} \frac{1-\chi^{\left(L_{U}(T)\right)}}{W_{U}(\mathbf{t})}$$
The coefficient of $1 / W_{U}(t)$ on the right hand side is
$$-\varepsilon(S) \sum_{U \subset T \subseteq S} \varepsilon(T)\left[1-\chi\left(L_{U}(T)\right)\right] .$$

## ECON346 COURSE NOTES ：

(Right-angled polygon groups.) Suppose $W$ is right angled with nerve a $k$-gon, $k \geqslant 4$, and that $\mathbf{t}$ is a single indeterminate $t$. Using Theorem as before, we get
$$\frac{1}{W(t)}=1-\frac{k t}{1+t}+\frac{k t^{2}}{(1+t)^{2}}=\frac{t^{2}+(2-k) t+1}{(1+t)^{2}} .$$
The roots of the numerator are $\rho$ and $\rho^{-1}$; so
$$\rho^{\pm 1}=\frac{(k-2) \mp \sqrt{k^{2}-4 k}}{2},$$
e.g., $\rho=\frac{3-\sqrt{5}}{2}$ when $k=5$.