About: Orientation sheaf

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In the mathematical field of algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at a point x is (in the integer coefficients or some other coefficients). Let be the sheaf of differential k-forms on a manifold M. If n is the dimension of M, then the sheaf is called the sheaf of (smooth) densities on M. The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map:

Attributes	Values
rdfs:label	Orientation sheaf (en)
rdfs:comment	In the mathematical field of algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at a point x is (in the integer coefficients or some other coefficients). Let be the sheaf of differential k-forms on a manifold M. If n is the dimension of M, then the sheaf is called the sheaf of (smooth) densities on M. The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map: (en)
dcterms:subject	Orientation (geometry) Algebraic topology
Wikipage page ID	49112440 (xsd:integer)
Wikipage revision ID	1067523932 (xsd:integer)
Link from a Wikipage to another Wikipage	Algebraic topology Orientation (geometry) Locally constant sheaf Algebraic topology Differential form Verdier duality Orientation of a manifold Integration of a differential form
Link from a Wikipage to an external page	https://mathoverflow.net/q/10966
sameAs	Orientation sheaf Orientation sheaf
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Topology-stub
has abstract	In the mathematical field of algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at a point x is (in the integer coefficients or some other coefficients). Let be the sheaf of differential k-forms on a manifold M. If n is the dimension of M, then the sheaf is called the sheaf of (smooth) densities on M. The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map: If M is oriented; i.e., the orientation sheaf of the tangent bundle of M is literally trivial, then the above reduces to the usual integration of a differential form. (en)
prov:wasDerivedFrom	wikipedia-en:Orientation_sheaf?oldid=1067523932&ns=0
page length (characters) of wiki page	1847 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Orientation_sheaf
is Link from a Wikipage to another Wikipage of	Algebraic analysis Orientation of a vector bundle Glossary of algebraic topology Locally constant sheaf Coherent sheaf cohomology Orientability Exceptional inverse image functor Sheaf cohomology Glossary of algebraic geometry Orientation bundle
is Wikipage redirect of	Orientation bundle
is foaf:primaryTopic of	wikipedia-en:Orientation_sheaf

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