The de Rham cohomology of soft function algebras

TL;DR

This paper investigates the algebraic de Rham cohomology of soft function algebras, revealing a canonical splitting and demonstrating nontrivial cohomology in certain cases, with implications for spaces like compact manifolds and polyhedra.

Contribution

It provides a canonical splitting of the de Rham cohomology for soft function algebras and explores specific cases including compact spaces and polyhedra, extending understanding of their algebraic topology.

Findings

Canonical splitting of cohomology for soft function algebras

Isomorphism between de Rham cohomology and singular cohomology for polyhedra

Nontrivial algebraic de Rham cohomology for infinite compact spaces

Abstract

We study the dg-algebra $\Omega ^\bullet_{A|\mathbb{R}}$ of algebraic de Rham forms of a real soft function algebra $A$, i.e., the algebra of global sections of a soft subsheaf of $C_X$, the sheaf of continuous functions on a space $X$. We obtain a canonical splitting $\mathrm H ^n (\Omega ^\bullet_{A|\mathbb{R}}) \cong \mathrm H ^n (X,\mathbb{R})\oplus V$, where $V$ is some vector space. In particular, we consider the cases $A=C(X)$ for $X$ a compact Hausdorff space and $A = C^\infty (X)$ for $X$ a compact smooth manifold. For the algebra $\mathrm{PPol}_K (|K|)$ of piecewise polynomial functions on a polyhedron $K$ the above splitting reduces to a canonical isomorphism $\mathrm H ^* (\Omega ^\bullet_{\mathrm{PPol}_K (|K|)|\mathbb{R}}) \cong \mathrm H ^* (|K|,\mathbb{R})$. We also prove that the algebraic de Rham cohomology $\mathrm H ^n (\Omega ^\bullet_{C(X)|\mathbb{R}})$ is nontrivial for each $n\geq 1$ if $X$ is an infinite compact Hausdorff space.