nLab real cohomology

Idea

By real cohomology one usually means ordinary cohomology with real number coefficients, denoted $H^\bullet\big(-, \mathbb{R}\big)$.

Hence, with the pertinent conditions on the domain space $X$ satisfied, its real cohomology $H^\bullet\big(-, \mathbb{R}\big)$ is what is computed by the Cech cohomology or singular cohomology or sheaf cohomology of $X$ with coefficients in $\mathbb{R}$.

In particuar, for $X$ a smooth manifold, the de Rham theorem says that real cohomology of $X$ is also computed by the de Rham cohomology of $X$

$$H^\bullet\big( X, \mathbb{R}\big) \;\simeq\; H^\bullet_{dR}\big( X \big) \,.$$

More generally, for $X$ a smooth manifold with smooth action of a connected compact Lie group, the equivariant de Rham theorem says that the real cohomology of the homotopy quotient (e.g. Borel construction) of $X$ is computed by the Cartan model for equivariant de Rham cohomology on $X$.

cohomology	Borel-equivariant cohomology
real ordinary cohomology	real equivariant ordinary cohomology
de Rham cohomology	equivariant de Rham cohomology

Properties

• de Rham theorem

• PL de Rham theorem

Related concepts

• real vector space

• integral cohomology, rational cohomology, complex cohomology

• de Rham cohomology, de Rham theorem

• equivariant de Rham cohomology, equivariant de Rham theorem

• rational homotopy theory

• rational Cohomotopy