A topological space \(X\) is acyclic if its reduced singular homology vanishes, \(\tilde H_n(X;\ZZ) = 0\) for all \(n\). A chain complex \((C_*, \del)\) is acyclic if \(H_n(C_*) = 0\) for all \(n\).
A nonempty space is acyclic if and only if the map to a point induces isomorphisms on all homology groups. A chain complex is acyclic if and only if it is exact.
[@Hat02].