# ag.algebraic geometry – Necessary and sufficient condition for the lifting of a proper map to be proper

For $$i=1,2$$ let $$(X_i,v_i)$$ be two connected topological manifolds, two subgroups $$H_ileq pi_1(X_i,v_i)$$, two coverings $$q_{H_i}:big(overline{X_i}(H_i),overline{v_i}big)to (X_i,v_i)$$ corresponding to $$H_i$$.

Consider a proper map $$f:(X_1,v_1)to (X_2,v_2)$$ such that $$f_#(H_1)leq H_2$$ i.e. we have a lift $$overline{f}:big(overline{X_1}(H_1),overline{v_1}big)to big(overline{X_2}(H_2),overline{v_2}big)$$ with $$q_{H_2}circoverline f=fcirc q_{H_1}$$.

$$require{AMScd}$$
$$begin{CD} left(overline{X_1}(H_1),overline{v_1}right) @>displaystyleoverline f>> left(overline{X_2}(H_2),overline{v_2}right)\ @Vq_{H_1} V V @VV q_{H_2}V\ left(X_1,v_1right) @>>displaystyle f> left(X_2,v_2right)\ end{CD}$$

Is there any necessary and sufficient condintions on $$f, H_1,H_2$$ such
that $$overline f$$ is a proper map?