|analytic integration||cohomological integration|
|measurable space||Poincaré duality|
|measure||orientation in generalized cohomology|
|volume form||(virtual) fundamental class|
|Riemann/Lebesgue integration of differential forms||push-forward in generalized cohomology/in differential cohomology|
For a manifold and a vector bundle of rank , an orientation on is an equivalence class of trivializations of the real line bundle that is obtained by associating to each fiber of its skew-symmetric th tensor power.
Equivalently for a smooth manifold this is an equivalence class of an everywhere non-vanishing element of .
An orientation of the tangent bundle or cotangent bundle is called an orientation of the manifold. This is equivalently a choice of no-where vanishing differential form on of degree the dimension of .
If a trivialization of exists, is called orientable.
For an orientation, is the opposite orientation.
From this perspective a choice of orientation is the first in a series of special structures on that continue with
For an E-∞ ring spectrum, tthere is a general notion of -orientation of vector bundles. This is described at
|smooth ∞-group||Whitehead tower of smooth moduli ∞-stacks||G-structure/higher spin structure||obstruction|
|ninebrane 10-group||ninebrane structure||third fractional Pontryagin class|
|fivebrane 6-group||fivebrane structure||second fractional Pontryagin class|
|string 2-group||string structure||first fractional Pontryagin class|
|spin group||spin structure||second Stiefel-Whitney class|
|special orthogonal group||orientation structure||first Stiefel-Whitney class|
|orthogonal group||orthogonal structure/vielbein/Riemannian metric|
|general linear group||smooth manifold|
(all hooks are homotopy fiber sequences)