cohomology(k, Delta, R)
Each abstract simplicial complex $\Delta$ determines a chain complex $\widetilde C(\Delta, k)$ of free modules over its coefficient ring $k$. For all integers $i$, the $i$-th term of $\widetilde C(\Delta, k)$ has a basis corresponding to the $i$-dimensional faces of $\Delta$. The reduced cohomology of $\Delta$ with coefficients in $R$ is, by definition, the cohomology of the chain complex $\operatorname{Hom}( \widetilde C(\Delta, k) \otimes R, R)$.
The 2-sphere has vanishing first cohomology, but non-trivial second cohomology.
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A figure eight has rank two first cohomology.
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