Cycle group

Definition. A chain is called a cycle if its boundary is $0$.

Theorem. The cycles form a subgroup $Z_k(K)$ of the chain group $C_k(K)$ which is the kernel of the boundary operator: $$Z_k(K) = {\rm ker}( \partial ).$$

This subgroup is called the cycle group ("$Z$" here stands for the word "cycle" in German).