Orientation sheaf

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources.
Find sources: "Orientation sheaf" – news · newspapers · books · scholar · JSTOR (April 2024)

|

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

o X , x = H n ⁡ ( X , X − { x } ) {\displaystyle o_{X,x}=\operatorname {H} _{n}(X,X-\\{x\\})}

(in the integer coefficients or some other coefficients).

Let  Ω M k {\displaystyle \Omega _{M}^{k}} be the sheaf of differential k-forms on a manifold M. If n is the dimension of M, then the sheaf

V M = Ω M n ⊗ o M {\displaystyle {\mathcal {V}}_{M}=\Omega _{M}^{n}\otimes {\mathcal {o}}_{M}}

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:

∫ M : Γ c ( M , V M ) -> R . {\displaystyle \textstyle \int _{M}:\Gamma _{c}(M,{\mathcal {V}}_{M})\to \mathbb {R} .}

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

[edit]

* 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

[edit]

* Kashiwara, Masaki; Schapira, Pierre (2002), Sheaves on Manifolds, Berlin: Springer, ISBN 3540518614

## External links

[edit]

* Two kinds of orientability/orientation for a differentiable manifold

This topology-related article is a stub. You can help Wikipedia by expanding it.

|

* v
* t
* e

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template