homology(Delta, R)
Each abstract simplicial complex $\Delta$ determines a chain complex of free modules over its coefficient ring. For all integers $i$, the $i$-th term in this chain complex has a basis corresponding to the $i$-th faces in the simplicial complex $\Delta$. When the optional argument $R$ is include, the chain complex is tensored with the given ring. The reduced homology of $\Delta$ with coefficients in $R$ is, by definition, the homology of this chain complex.
The $2$-sphere has vanishing first homology, but non-trivial second homology. We obtain a triangulation of the $2$-sphere by taking the $2$-skeleton of the $3$-simplex. Since homology groups are typically expressed as a subquotient, we prune the output to obtain a minimal presentation.
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The reduced homology of the Klein bottle has torsion.
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There are two "trivial" simplicial complexes: the irrelevant complex has the empty set as a facet whereas the void complex has no faces. Every abstract simplicial complex other than the void complex has a unique face of dimension $-1$.
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