and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
By rational cohomology one usually means ordinary cohomology with rational number coefficients, denoted $H^\bullet\big(-, \mathbb{Q}\big)$.
Hence, with the pertinent conditions on the domain space $X$ satisfied, its rational cohomology $H^\bullet\big(-, \mathbb{Q}\big)$ is what is computed by the Cech cohomology or singular cohomology or sheaf cohomology of $X$ with coefficients in $\mathbb{Q}$.
(universal coefficient theorem in rational cohomology)
For rational numbers-coefficients $\mathbb{Q}$, the Ext groups $Ext^1(-;\mathbb{Q})$ vanish, and hence the universal coefficient theorem identifies rational cohomology groups with the dual vector space of the rational vector space of rational homology groups:
(e.g. Moerman 15, Cor. 1.2.1)
Last revised on March 9, 2021 at 03:43:30. See the history of this page for a list of all contributions to it.