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 $$ -\varepsilon(S) \sum_{U \subset T \subseteq S} \varepsilon(T)\left[1-\chi\left(L_{U}(T)\right)\right] . $$
(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 $k=5$.
Suppose a virrually sorsiont-free group $G$ acts properly and cocompactly on an acyclic conplex $Y$ whose cohonology wibh connpact supports is given by $$ H_{e}^{i}(Y) \cong \begin{cases}0 & \text { if } i \neq n \ Z & \text { if } i=n\end{cases} $$ Then $G$ is a virual $P^{n}$-group.
证明 .
Since $Y / G$ is compact, $G$ is type $V F L$. By Lemma F.2.2, $$ H_{c}^{i}(G, Z G) \cong H_{c}^{i}(Y) \cong \begin{cases}0 & \text { if } i \neq n, \ Z & \text { if } i=n .\end{cases} $$ and the same formula holds for any torsion-free subgroup $\pi$ of finite index in $G$.
where $a=5<n_{1}<\cdots I_{n}=b$ runs over all possible subdivisions of $[a, b]$. $(X, d)$ is a length grace if $$ d(x, y)=\inf {\Omega(\gamma) \mid \gamma \text { is a path from } x \text { to } y} . $$ (Here we allow oo as a possible value of $d$.) Thus, a length space is a geodesic space if the above infimum is alw ays realized and is $\neq \infty$.
Suppose $\pi$ is a group and that $B \pi$ is its classifying space. As is usual, the universal cover of $B \pi$ is denoted $E \pi$. ( $E \pi$ is called the universal space for $\pi$.) Given a $\mathbb{Z} \pi$-module $A$, the homology and cohomology groups of $\pi$ with coefficients in $A$ are defined by $$ H_{}(\pi ; A):=H_{}(B \pi ; A) \quad \text { and } \quad H^{}(\pi ; A):=H^{}(B \pi ; A) . $$
证明 .
Let $\varepsilon: C_{0}(E \pi) \rightarrow \mathbb{Z}$ be the augmentation map. Since $E \pi$ is acyclic (it is contractible), the sequence $$ \longrightarrow C_{k}(E \pi) \longrightarrow \cdots \longrightarrow C_{0}(E \pi) \stackrel{\varepsilon}{\longrightarrow} \mathbb{Z} \longrightarrow 0 $$ is exact. In other words, (F.4) is a resolution of $\mathbb{Z}$ by free $\mathbb{Z} \pi$-modules. (Here and throughout the group $\pi$ acts trivially on $\mathbb{Z}$.)
$$ \operatorname{Hom}{G}\left(\mathbb{Z}\left(G / G{\sigma}\right), \operatorname{Hom}\left(F_{}, \mathbb{Z} G\right)\right) \cong \operatorname{Hom}{G{\sigma}}\left(F_{}, \mathbb{Z} G\right) .^{3} $$ The cohomology of this last complex is just $H^{q}\left(G_{\sigma} ; \mathbb{Z} G\right)$. Taking cohomology first with respect to $q$ we get a spectral sequence whose $E_{1}^{p q}$ term is a sum of terms of the form $H^{q}\left(G_{\sigma} ; \mathbb{Z} G\right)$. Since $\mathbb{Z} G$ is a free $G_{\sigma}$-module and since $G_{\sigma}$ is finite (because the action is proper), these groups vanish for $q>0$. For $q=0$ they are the invariants, $(\mathbb{Z} G)^{G_{\sigma}}$. All that remains is $$ E_{1}^{p, 0}=\bigoplus(\mathbb{Z} G)^{G_{\sigma}} \cong \operatorname{Hom}{G}\left(C{p}, \mathbb{Z} G\right)=C_{G}^{p}(Y) $$ So $E_{2}^{p, 0} \cong H_{G}^{p}(Y ; \mathbb{Z} G)$. Since the $G$-action on $Y$ is cocompact, Lemma F.2.1 (i) gives $H_{G}^{p}(Y ; \mathbb{Z} G)=H_{c}^{p}(Y)$, completing the proof.
Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron. This led to his polyhedron formula, V − E + F = 2 (where V, E, and F respectively indicate the number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology.
拓扑学课后作业代写
(i) Suppose $W$ is finite and $t_{S}$ is the monomial corresponding to its element of longest length. Then $$ t_{S}=W(\mathbf{t}) \sum_{T \subset S} \frac{\varepsilon(T)}{W_{T}(\mathbf{t})} . $$ (ii) If $W$ is infinite, then $$ 0=\sum_{T \subset S} \frac{\varepsilon(T)}{W_{T}(\mathbf{t})} $$ Proof. Apply Lemma 17.1.4 in the case $T=S$, to get $$ W^{s}(\mathbf{t})=W(\mathbf{t}) \sum_{U \subset S} \frac{\varepsilon(S-U)}{W_{S-U}(\mathbf{t})} $$ Suppose $W$ is finite. Then, by Lemma $4.6 .1, W^{S}=\left{w_{S}\right}$; so, $W^{S}(\mathbf{t})=t_{S}$. Reindex the sum by setting $T=S-U$, to get (i).
If $W$ is infinite, then $W^{S}=\emptyset$ and $W^{S}(\mathbf{t})=0$. Reindex the sum by setting $T=S-U$ and then divide by $W(\mathrm{t})$ to get (ii).
