{{one source |date=April 2024}} In the mathematical field of algebraic topology, the '''orientation sheaf''' on a manifold ''X''<!-- or some reasonable f-dim space to which there is duality. --> of dimension ''n'' is a locally constant sheaf ''o''<sub>''X''</sub> on ''X'' such that the stalk of ''o''<sub>''X''</sub> at a point ''x'' is the local homology group :<math>o_{X, x} = \operatorname{H}_n(X, X - \{x\})</math> (in the integer coefficients or some other coefficients).
Let <math>\Omega^k_M</math> be the sheaf of differential ''k''-forms on a manifold ''M''. If ''n'' is the dimension of ''M'', then the sheaf :<math>\mathcal{V}_M = \Omega^n_M \otimes \mathcal{o}_M</math> 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: :<math>\textstyle \int_M: \Gamma_c(M, \mathcal{V}_M) \to \mathbb{R}.</math> 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.
== See also == *There is also a definition in terms of dualizing complex in Verdier duality; in particular, one can define a relative orientation sheaf using a relative dualizing complex.
== References == *{{Citation | last1=Kashiwara | first1=Masaki | last2=Schapira | first2=Pierre | author1-link=Masaki Kashiwara | title=Sheaves on Manifolds | isbn=3540518614 | year=2002 | publisher=Springer | location=Berlin}}
== External links == *[https://mathoverflow.net/q/10966 Two kinds of orientability/orientation for a differentiable manifold]
Category:Algebraic topology Category:Orientation (geometry)
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