Prove that two cycles that surround the same holes differ by a boundary i.e. the relation for calling two cycles homologous as mentioned here.
2026-05-04 21:37:42.1777930662
Proof for Homologous cycles
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What it means to "surround the same hole" is that the two cycles are in the same homology class (it also depends on the multiplicity, not just which holes are surrounded), and so by definition their difference is a boundary.
I think what you're looking to do is prove that homology is well-defined. This is not a trivial task, but the standard reference is Chapter 2 of Hatcher's Algebraic Topology.
The idea, however, is the same as the idea of fundamental groups. You can go around one cycle in one direction, then the other in the reverse direction. This is a "closed loop" whose interior doesn't include the hole, and is hence a boundary.