Orientation sheaf
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
- [math]\displaystyle{ o_{X, x} = \operatorname{H}_n(X, X - \{x\}) }[/math]
(in the integer coefficients or some other coefficients).
Let [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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
- 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
External links
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