group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
| analytic integration | cohomological integration |
|---|---|
| measure | orientation in generalized cohomology |
| Riemann/Lebesgue integration, of differential forms | push-forward in generalized cohomology/in differential cohomology |
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
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.
An orientation on a Riemannian manifold is equivalently a lift of the classifying map of its tangent bundle through the fist step in the Whitehead tower of :
From this perspective a choice of orientation is the first in a series of special structures on that continue with
orientation
For an E-∞ ring spectrum, tthere is a general notion of -orientation of vector bundles. This is described at
For be the Eilenberg-MacLane spectrum for the discrete abelian group of real numbers, orientation in -cohomology is equivalent to the ordinary notion of orientation described above.
(all hooks are homotopy fiber sequences)