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

TL;DR

The paper explicitly describes the isomorphism between singular and Čech cohomology as evaluation on the Čech nerve, clarifying the relationship through a homotopy equivalence.

Contribution

It provides an explicit description of the unique isomorphism between singular and Čech cohomology via evaluation on the Čech nerve.

Findings

Explicit isomorphism between singular and Čech cohomology.

Evaluation at simplices of the Čech nerve characterizes the isomorphism.

Clarifies the relationship via a homotopy equivalence.

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 $N\mathcal{U}\to X$.