Uintell Mathematics Encyclopedia

cycle of index zero

A cycle $z=\sum_{i=1}^na_it_i^0$ such that $\sum_{i=1}^na_i=0$. A cycle homologous to zero is always normal; the quotient group of the group of normal cycles by that of the cycles homologous to zero is called the reduced zero-dimensional homology group. For a connected complex the reduced group is zero, which is convenient in any kind of definition of acyclicity. A proper cycle $z^0=\{z_1^0,\dots,z_k^0,\dots\}$ is called normal if each of the cycles $z_k^0$ is normal.