The evaluation isomorphism of singular cohomology on the \v{C}ech nerve

Abstract

We show that the canonical isomorphism $\mathrm{H}_{sin}^*(X,\underline{\mathbb{Z}})\xrightarrow{\sim} \check{\mathrm{H}}_{\mathcal{U}}^*(X,\underline{\mathbb{Z}})$ is the evaluation of singular cohomology classes at the simplices of the \v{C}ech nerve $ N\mathcal{U}$ through a homotopy equivalence $N\mathcal{U}\to X$. We apply this to real tori $V/\Lambda$ to show that the automorphy map $\mathrm{H}_{sin}^*(V/\Lambda,\mathbb{Z})\xrightarrow{\sim} \mathrm{H}^*(\Lambda,\mathbb{Z})$ and the group cohomology map $ \mathrm{H}^*(\Lambda,\mathbb{Z}) \xrightarrow{\sim} \check{\mathrm{H}}_{\mathcal{U}}^*(V/\Lambda,\underline{\mathbb{Z}}) $ induced by the universal cover are inverse to each other.