Skip to main content
2 of 2
added 16 characters in body
David E Speyer
  • 156.3k
  • 14
  • 421
  • 763

A topological group is homogenous. In a finite CW complex, the interior of every top dimensional cell is a topological manifold. Therefore, a finite CW complex which is a toplogical group is a topological manifold. By Hilbert's fifth problem,a topological group which is a topological manifold is a Lie group (not necessarily connected).

Finite CW complexes are also compact, so your group is a compact Lie group.

Positive dimensional Lie groups have torus subgroups, hence have $\chi(G)=0$, so $BG$ is not a finite CW complex as you say.

Zero dimensional Lie groups have $\chi(G) = |G|$, so $BG$ is only a finite CW complex if $|G|=1$.

David E Speyer
  • 156.3k
  • 14
  • 421
  • 763