Recall that discrete lattice groups Z k always acts freely on R k pointwise, and the quotient space R k /Z k = T k , is still a manifold. The simple example resulting from these contents will be listed in next section. Meanwhile, any finite group cannot act freely on R k from Smith theory of group actions. Proposition 5.1 (Persistent homology of covering space). If F is a Γ-invariant filtration acting on K, with K l as its normal covering space for any K l /Γ. Suppose F be a field whose charateristic is comprime to |Γ|, then H l,p k (K) = 0 if and only if H l,p k (K/Γ) = 0. That is to say, the transfer homomorphisms of persistent homology are isomorphisms for trivial homology groups.