In this article, it is agreed that the following terms and notations can be denoted for convenience purposes: X is a metrizable and triangulable topological space, G for any group, Γ for any discrete group (or a crystallographic group), and K for any CW G -complex, which means that with any celluar G-action, if g ∈ G fixes a cell σ ∈ K, then g fixes σ poitwise. Hence, one can clearly deduct that there is a set of fixed points K G which is a subcomplex of K. The quotient space K/G or K/Γ derived from covering mapping is clearly a CW -complex. For brevity, we always apply complex to denote CW-complex or Cell complex unless otherwise stated. One can consider any ring R or abelian groups A as coefficient for chain complexes. one can omit the notation for coefficients for chain complexes since the universal coefficient theorem always holds. To make it practical, we restrict our consideration only to discrete groups rather than generalizing to compact Lie groups. Following conventional notation in persistent homology theory, a filtration of a complex K is a mapping {φ : K×N → K} ordered sequences of subcomplexes, Ø =