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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, the orientation sheaf on a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
''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 the
local homology group
:
(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
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
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
*
External links
Two kinds of orientability/orientation for a differentiable manifold
Algebraic topology
Orientation (geometry)
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