Consider the universal bundle $G\hookrightarrow EG\rightarrow BG$. Is it possible to get another bundle $EG_{B}$ by restricting $EG$ over a smooth singular $k$chain $B \in H_k(BG)$ ($k\leq n$)? I know it's perfectly fine to restrict $EG$ to a submanifold of $BG$, but what about chains?
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If you have a $G$bundle over $M$ (without boundary) then this corresponds to the homotopy class of a map $\gamma:M\to BG$. It is known that for $G$ simply connected, any $G$bundle over $M$ is trivializable. One way to see this is by obstruction theory. The other is to notice that $\pi_i(BG)=\pi_{i1}(G)$ and for $i=1,2,3$ this is zero. Therefore, one can get a cellular model for $BG$ which has no $3$cells. Therefore, $\gamma$ is homotopic to a cellular map $\gamma':M\to BG$ which is necessarily constant. Since $\gamma_\ast[M]=0$ in $H_3(BG)$ by Hurewicz and the previous observations, there is a singular $4$chain $B$ with boundary $\gamma_\ast[M]$. Then using the necessary pullbacks you get what you want. 

