# Orientation sheaf

In algebraic topology, the **orientation sheaf** on a manifold *X* of dimension *n* is a locally constant sheaf *o*_{X} on *X* such that the stalk of *o*_{X} 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.

## See also

- Orientation of a manifold
- 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

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

This article is issued from
Wikipedia.
The text is licensed under Creative
Commons - Attribution - Sharealike.
Additional terms may apply for the media files.