In the mathematical field of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the orientation sheaf on a manifold ''X'' of dimension ''n'' is a
locally constant sheaf In algebraic topology, a locally constant sheaf on a topological space ''X'' is a sheaf \mathcal on ''X'' such that for each ''x'' in ''X'', there is an open neighborhood ''U'' of ''x'' such that the restriction \mathcal, _U is a constant sheaf on ...
''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 mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of
Alexander Grot ...
; in particular, one can define a relative orientation sheaf using a relative dualizing complex.
References
*
External links
Two kinds of orientability/orientation for a differentiable manifold
{{topology-stub
Algebraic topology
Orientation (geometry)