Zero-homologous cycle

Definition

Let $U\subset \mathbb {C}$ be an open subset. A cycle $c$ in $U$ is termed zero-homologous or nullhomologous if it satisfies the following equivalent conditions:

• Its homology class in the homology group $H_{1}(U)$, is zero
• $c$ has zero winding number around any point in $\mathbb {C} \setminus U$
• $c$ does not separate the complement of $U$