If $G \neq \lbrace 1 \rbrace$ is a finite group with classifying space $BG$ then there are infinitely many i such that $H^i(BG,\mathbb{Z}) \neq 0$. This can be found, for example, there:

Non-vanishing of group cohomology in sufficiently high degree

As a consequence, the CW-complex $BG$ (unique up to homotopy) can not be of finite dimension.

Question: Are there alternative proofs for this observation. In particular, I would be interested in knowing if there is a purely topological proof without homological algebra.

If $G \neq \lbrace 1 \rbrace$ is a finite group with classifying space $BG$ then there are infinitely many i such that $H^i(BG,\mathbb{Z}) \neq 0$. This can be found, for example, there:

Non-vanishing of group cohomology in sufficiently high degree

As a consequence, the CW-complex $BG$ (unique up to homotopy) can not be of finite dimension.

Question: Are there alternative proofs for this observation. In particular, I would be interested in knowing if there is a purely topological proof without homological algebra.