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  "body": "\n## Summary\n\nSimplicial homology assigns to a [[simplicial complex|V8GZ0W1ZZ5]] a sequence of abelian groups $H_n(K)$ that count its $n$-dimensional holes.\nFor a finite complex it is computed by finite linear algebra over $\\Z$.\nChains are formal sums of oriented simplices, and the boundary operator sends a simplex to the signed sum of its faces.\nIt agrees with the [[singular homology|M0Q5CFJ7ZH]] of the underlying space, so it is a topological invariant even though its definition uses a triangulation.\n\n## Explanations\n\n### L1 human\n\nA cycle is a combination of simplices with no boundary, like a loop of edges.\nA boundary is a cycle that is filled in by higher simplices, like the edge loop around a triangle that is present in the complex.\nThe $n$-th homology group counts cycles that are not boundaries.\nFor the hollow triangle, made of three edges, the loop around it is a cycle that is not filled, so $H_1 \\cong \\Z$.\nIf the triangle is filled in, the loop becomes a boundary and $H_1 = 0$.\n",
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