de Rham cohomology

E551975

cohomology theory functor topological invariant

de Rham cohomology is a cohomology theory for smooth manifolds that uses differential forms to capture their global topological and geometric properties.

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Observed surface forms (1)

Surface form	Occurrences
de Rham theorem	1

Statements (49)

Predicate	Object
instanceOf	cohomology theory ⓘ functor ⓘ topological invariant ⓘ
appliesTo	non-compact smooth manifolds ⓘ oriented smooth manifolds ⓘ
captures	global geometric properties of manifolds ⓘ global topological properties of manifolds ⓘ
cochainComplex	(Ω^*(M), d) ⓘ
coefficientsCanBe	complex numbers ⓘ
coefficientsTypically	real numbers ⓘ
cohomologyGroupNotation	H^k_{dR}(M) ⓘ
constructedFrom	complex of differential forms ⓘ exterior derivative ⓘ
definedAs	cohomology of the de Rham complex ⓘ
domain	smooth manifolds ⓘ
field	algebraic topology ⓘ differential geometry ⓘ global analysis ⓘ
generalizes	notion of closed and exact differential forms ⓘ
H0Interpretation	space of locally constant functions on a connected manifold ⓘ
historicalPeriod	20th century mathematics ⓘ
HnInterpretation	top-degree cohomology related to orientation and volume forms ⓘ
invariantUnder	diffeomorphisms of manifolds ⓘ
isFunctorFrom	category of smooth manifolds with smooth maps ⓘ
isFunctorTo	category of graded real vector spaces ⓘ
isomorphicTo	singular cohomology with real coefficients for smooth manifolds ⓘ
isomorphismType	H^k_{dR}(M) ≅ H^k_{sing}(M;ℝ) ⓘ
kthGroupDefinition	kernel of d:Ω^k→Ω^{k+1} modulo image of d:Ω^{k-1}→Ω^k ⓘ
namedAfter	Georges de Rham NERFINISHED ⓘ
relatedConcept	Hodge theory NERFINISHED ⓘ Mayer–Vietoris sequence NERFINISHED ⓘ Poincaré duality NERFINISHED ⓘ Poincaré lemma NERFINISHED ⓘ elliptic complexes ⓘ sheaf cohomology ⓘ Čech–de Rham complex NERFINISHED ⓘ
relatedTo	singular cohomology ⓘ
satisfies	de Rham theorem NERFINISHED ⓘ
toolFor	classification of smooth manifolds up to homotopy type ⓘ formulation of Stokes theorem in cohomological terms ⓘ study of integration of differential forms ⓘ
usedIn	gauge theory ⓘ index theory ⓘ string theory NERFINISHED ⓘ theory of characteristic classes ⓘ
uses	differential forms ⓘ
variant	compactly supported de Rham cohomology ⓘ equivariant de Rham cohomology ⓘ relative de Rham cohomology ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Poincaré lemma → dealsWith → de Rham cohomology ⓘ

Weil cohomology → hasExample → de Rham cohomology ⓘ

Deligne cohomology → refines → de Rham cohomology ⓘ

Hodge theory → relatedTo → de Rham cohomology ⓘ

Chern character → relatesTo → de Rham cohomology ⓘ

Poincaré lemma → usedInProofOf → de Rham cohomology ⓘ

this entity surface form: de Rham theorem

Chern–Weil theory → usesConcept → de Rham cohomology ⓘ