Ths section examines some validity of covering mapping on persistent homology groups, which leads to the most important results in Theorem 5.2, that covering spaces will share the same bar code diagrams with their quotient spaces. Some transfer homomorphism will be constructed more precisely in Theorem 5.1 & 5.3. Persistent homology groups are derived form homology over subspaces series of filtration, where F(K) is a filtration on manifolds M. Then, Z l k = Z k (K l ) and B l k = B k (K l ) are the k-th cycle and boundary groups of subcomplex K l ⊆ K respectively.The k-th homology group of