 Null-homologous cycle

A chain cycle is null-homologous with the chosen coefficients when its homology class is zero. Equivalently it is a chain boundary: there is a chain $b$ one degree higher such that $\partial b=c$.

A geometric bounding argument over the integers needs compatible orientations. If a compact oriented submanifold $W$ has oriented boundary $C$, a chain representing its relative fundamental class has boundary representing $C$, so $[C]=0$ in the ambient integral homology. With mod-two coefficients, orientation signs disappear and an unoriented bounding submanifold suffices.

For example, the boundary circle of a Möbius band bounds that band as an unoriented surface, but represents twice the core generator in $H_1(\text{Mo¨bius band};\mathbb{Z})\cong \mathbb{Z}$. It is not integrally null-homologous inside the band. Its mod-two homology class is zero. This distinguishes bounding a geometric subset from bounding a chain with the required coefficients.