The isomorphism to Čech cohomology as evaluation on the Čech nerve

TL;DR

The paper establishes an explicit isomorphism between integral singular cohomology and Čech cohomology for a good cover by evaluating singular cochains on the Čech nerve, using a homotopy equivalence that respects the cover. It reduces the problem to a saturated cover, transfers to a $\Delta$-complex model, and constructs a cone operator to realize the diagram chase as an evaluation on the nerve, with a precise sign $(-1)^{k(k+1)/2}$. This yields a concrete, computable description of the isomorphism and connects the integral theory to a de Rham-type realization via the $\Delta$-complex, clarifying the relationship between different cohomological formalisms. The discussion also addresses the role of the sign and proposes a natural combinatorial model based on $\mathrm{H}_{\Delta}^*(N\mathcal{U},\mathbb{Z})$ (with evaluation), highlighting implications for both topology and algebraic geometry.

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 Čech acyclic sheaf resolutions as the evaluation, up to sign, of singular cohomology classes at the simplices of the Čech nerve $N\mathcal{U}$ through a homotopy equivalence $N\mathcal{U}\to X$.

Figures (1)

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Theorems & Definitions (14)

• Definition 1
• Lemma 2
• Theorem 3
• Remark 4
• Definition 5
• Definition 6
• Definition 7
• Lemma 8
• proof
• Lemma 9