= Null-homologous cycle
{title2=$[C]=0$}
= Null-homologous
{synonym}
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 $H_1(\text{Möbius band};\mathbb Z)\cong\mathbb Z$. 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.
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