As [pointed out](https://mathoverflow.net/posts/comments/1215521) by Achim Krause, any finite CW complex is homotopy equivalent to a manifold with boundary (by embedding into $\mathbb R^n$ and thickening), and so every finite type CW complex can be approximated by manifolds with boundary. Given a Lie group $G$, is it possible to approximate $BG$ by using boundaryless manifolds? For example, $BS^1$ can be approximated by $\mathbb{CP}^N$. Is this a general phenomenon or special case for torus? (The classifying space of a torus supports a group structure, so it has to be boundaryless.)