这是一份leeds利兹大学MATH322501作业代写的成功案例
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.
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 $G$.
MATH322501 COURSE NOTES ：
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 $\neq \infty$.