Now, we can construct some gernalization of our consideration for transfer homomorphism on persistent homology groups h # : H l,p k (K, F) → H l,p k (K/Γ, F) rather than only for zero spaces. We also abandon the restraint that F should be a field whose charateristic is either zero or comprime to |Γ|, in order to consider any coefficient group: Proposition 5.2. The transfer homomorphism on persistent homology groups, h # , is an isomorphism if and only if (ker p # ∩ Z l k (K)) ⊂ (ker p # ∩ B l k (K)). Proof. Clearly, one can consider an intersection diagram of spanning space of basis, since every subspace here is closed. Then, suppose H l,p k (K, F) ∼ = H l,p k (K/Γ, F), which leads to a vanishing kernel of h # . As known that ker h # = (Z l k (K) ∩ ker p # )\B l+p k (K), which is equivalent to (ker p # ∩ Z l k (K)) ⊂ (ker p # ∩ B l k (K)). More specefically, for free actions where local stabilizer group si always trivial, we have the following explicit expression. If |Γ| is finite and Γ act on X freely,which means that ∀x ∈ K, ∃gx = x ⇔ g = Id, then summation only contains a single term, indicating χ(X/Γ) = χ(X)/|Γ|. Corollary 5.2.1. If Γ acts freely on K and let coefficient group F be a field whose charateristic is comprime to |Γ|, there is: