A result of Manning states that for a compact manifold X and a continuous map \(f: X \rightarrow X\) the topological entropy of f is bounded below by the logarithm of the spectral radius of the map induced by f in the first homology group \(H_1(X;\mathbb {C})\) . We generalize this result to arbitrary compact spaces X in terms of Čech cohomology \(\check{H}^1(X;\mathbb {C})\) . The essential tool is a notion of integration of Alexander-Spanier cocycles over Čech cycles. Most of the discussion is carried out “at scale \(\mathcal {U}\) ”, for an open covering \(\mathcal {U}\) . This is used to keep track of the “homological length” of the iterates of a cycle, which in turn leads to a lower bound on the number of elements in the coverings that appear in the definition of the entropy.
