nLab
real cohomology

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Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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

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

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

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

More generally, for XX 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 XX is computed by the Cartan model for equivariant de Rham cohomology on XX.

cohomologyBorel-equivariant cohomology
real ordinary cohomologyreal equivariant ordinary cohomology
de Rham cohomologyequivariant de Rham cohomology

Last revised on June 27, 2019 at 09:19:40. See the history of this page for a list of all contributions to it.