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I came across this term "non-bounding" cycle in the context of homology. However I am not exactly sure what it means.

What I know is that cycles are element of $\ker\partial_k$ and boundaries are elements of $im \partial_{k+1}$. So my guess is that non-bounding cycle is a cycle that is not a boundary?

Thanks for any help.

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Examples of non-bounding cycles are toroidal or poloidal closed loops on a torus that do not enclose any area. Compare with closed loops on a spere all of which are bounding cycles.

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Back to the definitions, with $\mathbf{Z}_k=\ker \partial_k$ the group of k-cycles and $\mathbf{B}_k= \text{im }\partial_{k+1}$ the group of k-boundaries. As $\partial_k\circ\partial_{k+1}=0$ we have $\mathbf{B}_k\subset\mathbf{Z}_k$.

A k-cycle $c_k$ (by definition of cycles $c_k\in\mathbf{Z}_k$) is said non-boundary if $c_k\notin\mathbf{B}_k$