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