The isomorphism to \v{C}ech cohomology as evaluation on the \v{C}ech nerve

Abstract

We describe explicitely the unique isomorphism $\mathrm{H}_{sin}^*(X,\underline{\mathbb{Z}})\xrightarrow{\sim} \check{\mathrm{H}}_{\mathcal{U}}^*(X,\underline{\mathbb{Z}})$ between the cohomologies computed with the singular and \v{C}ech acyclic sheaf resolutions as the evaluation, up to sign, of singular cohomology classes at the simplices of the \v{C}ech nerve $N\mathcal{U}$ through a homotopy equivalence $N\mathcal{U}\to X$.