group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
For $X$ a manifold and $V \to X$ a vector bundle of rank $k$, an orientation on $V$ is an equivalence class of trivializations of the real line bundle $\wedge^k V$ that is obtained by associating to each fiber of $V$ its skew-symmetric $k$th tensor power.
Equivalently for a smooth manifold this is an equivalence class of an everywhere non-vanishing element of $\wedge^k_{C^\infty(X)} \Gamma(V)$.
An orientation of the tangent bundle $T X$ or cotangent bundle $T^* X$ is called an orientation of the manifold. This is equivalently a choice of no-where vanishing differential form on $X$ of degree the dimension of $X$.
If a trivialization of $\wedge^k V$ exists, $V$ is called orientable.
For $\omega$ an orientation, $-\omega$ is the opposite orientation.
An orientation on a Riemannian manifold $X$ is equivalently a lift $\hat g$ of the classifying map $g : X \to \mathcal{B}O(n)$ of its tangent bundle through the fist step $S O(n) \to O(n)$ in the Whitehead tower of $X$:
From this perspective a choice of orientation is the first in a series of special structures on $X$ that continue with
orientation
For $E$ an E-∞ ring spectrum, tthere is a general notion of $R$-orientation of vector bundles. This is described at
For $R = H(\mathbb{R})$ be the Eilenberg-MacLane spectrum for the discrete abelian group $\mathbb{R}$ of real numbers, orientation in $R$-cohomology is equivalent to the ordinary notion of orientation described above.
smooth ∞-group | Whitehead tower of smooth moduli ∞-stacks | G-structure/higher spin structure | obstruction |
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$\vdots$ | |||
$\downarrow$ | |||
fivebrane 6-group | $\mathbf{B}Fivebrane$ | fivebrane structure | second fractional Pontryagin class |
$\downarrow$ | |||
string 2-group | $\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$ | string structure | first fractional Pontryagin class |
$\downarrow$ | |||
spin group | $\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$ | spin structure | second Stiefel-Whitney class |
$\downarrow$ | |||
special orthogonal group | $\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$ | orientation structure | first Stiefel-Whitney class |
$\downarrow$ | |||
orthogonal group | $\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2$ | orthogonal structure/vielbein/Riemannian metric | |
$\downarrow$ | |||
general linear group | $\mathbf{B}GL$ | smooth manifold |
(all hooks are homotopy fiber sequences)