Suppose $\Gamma$ is a finitely presented group that has a torsion element. Can the classifying space $K(\Gamma,1)$ be homotopic to a finite-dimensional manifold?
If yes, what is the simplest example?



Classifying spaces of finitely presented groups with torsion elements
This is not possible. Let $B$ be a space of type $K(\Gamma,1)$ and of dimension $d<\infty$, and let $E$ be the universal cover, which is contractible. Then for any $A\leq\Gamma$ we see that $E/A$ is a $K(A,1)$ and has finite dimension, so the group cohomology $H^i(A)$ vanishes for $i>d$. However, for $n>1$ the group cohomology of $C\_n$ is nonzero in all even degrees, so $\Gamma$ cannot contain $C\_n$, so $\Gamma$ must be torsion-free. 


