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. Theorem 5.2 shows the sufficient condition allowing ones to calculate birth-death persistence on quotient spaces as birth-death time on barcodes or persistence diagrams is always invariant thus.