Consider a group $G$. We want to determine the universal space $EG$. Is it true that all universal spaces are homotopy equivalent. That is, to find $EG$ we only need to find a weakly contractible space $X$ on which $G$ acts freely and this will be necessarily our $EG$ because if there is another weakly contractible space $Y$ on which $G$ acts freely then $Y$ must be homotopy equivalent to $X$. Is my understanding correct or there is one specific weakly contractible space on which $G$ acts freely that we will take as our $EG$?
Another question is regarding when $G$ is discrete, we know that $BG=K(G,1)$ and in many situations there is a canonical way how to construct a $K(G,1)$ but how to determine $EG$ knowing $BG$? I mean if we know $EG$ we can determine $BG$ by taking the quotient $BG=EG/G$ but if we know $BG$ like in the discrete case how will we determine $EG$? thanks for your help!!
Given a discrete group $G$, any two $BG$'s are homotopy equivalent to each other. Also, we can simply construct $EG$ to be the universal cover of a $BG$. Also, every $EG$ is constructed in this manner. Also, since $BG$ is a CW complex, so is $EG$.
So no, there is not one specific $EG$, because there is not one specific $BG$. However, the $BG$'s do all vary within one specific homotopy equivalence class of CW-complexes.
Also, any weakly contractible CW-complex is contractible, and therefore all $EG$'s are contractible, so they are all homotopy equivalent to a point and therefore homotopy equivalent to each other, independent of the group $G$.