Hence, quotient space can always be given a Riemannian metric induced from covering map of isometry, then function F(K/G) is clearly induced from F(K) by isometric covering, which leads to the commutative diagram above. Subsequently, we consider the relation between the covering map p and the boundary operator ∂ in homology computation. In this case, a commutative diagram can be constructed that relates the boundary operator ∂ to the induced quotient mapping p # . Proposition 4.2. Suppose that K l ∈ F(K) and p is a covering map, there always exists a linear operator p i # satisfying p i # • S = S • p, where S is natrual spanning into linear chain space: