Universal covers of two non-homeomorphic $K(G,1)$ space

Question

I know any two $K(G,1)$ spaces are homotopic and their universal covers are contractible. I am just curious how universal covers of two $K(G,1)$ spaces are related.

Is there any general result like, when are they homeomorphic?

Answer 1

In fact, there is no hope for two $K(G,1)$'s being homeomorphic and similarly for their universal covers. Generally, for a topological space $G$ you could construct a universal bundle $EG \to BG$ where $BG = K(G,1)$ is a classifying space of $G$-principal bundles, means $[X,BG] = \text{Prin}_{G}(X)$ and $EG$ can be supposed to be spherical. In the category of connected CW-complex, it means $EG$ is homotopic to one point.

The universal property of this kind of bundle forces all $BG$'s to be unique up to a homotopy equivalence through a commutative diagram but no hope for $EG$'s. $\require{AMScd}$ \begin{CD} E_{1} @>{\widetilde{h}}>> E_{2}\\ @VVV @VVV\\ B_{1} @>{h}>> B_{2} \end{CD}