The notation d_n: C_n \to C_n-1 represents a boundary operator in algebraic topology, specifically in the context of chain complexes. Here, C_n denotes the group of n -chains, which are formal sums of n -dimensional simplices, while C_n-1 represents the group of (n-1) -chains. The operator d_n maps each n -chain to its boundary, which consists of the (n-1) -dimensional faces of the simplex. In this framework, the boundary operator is crucial for understanding the relationships between different dimensions of chains. It helps in defining homology groups, which are used to study topological spaces. The properties of d_n ensure that applying it twice results in zero, meaning d_n \circ d_n+1 = 0 . This property is