group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hirzebruch signature theorem?
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
Generally, for $E$ an E-∞ ring spectrum, and $P \to X$ a sphere spectrum-bundle, an $E$-orientation of $P$ is a trivialization of the associated $E$-bundle.
Specifically, for $P = Th(V)$ the Thom space of a vector bundle $V \to X$, an $E$-orientation of $V$ is an $E$-orientation of $P$.
More generally, for $A$ an $E$-algebra spectrum, an $E$-bundle is $A$-orientable if the associated $A$-bundle is trivializable. For more on this see (∞,1)-vector bundle.
The existence of an $E$-orientation is necessary in order to have a notion of fiber integration in $E$-cohomology.
Let $E$ be a multiplicative cohomology theory and let $V \to X$ be a topological vector bundle of rank $n$. Then an $E$-orientation or $E$-Thom class on $V$ is an element of degree $n$
in the reduced $E$-cohomology ring of the Thom space of $V$, such that for every point $x \in X$ its restriction $i_x^* u$ along
(for $\mathbb{R}^n \overset{fib_x}{\hookrightarrow} V$ the fiber of $V$ over $x$) is a generator, in that it is of the form
for
$\epsilon \in \tilde E^0(S^0)$ a unit in $E^\bullet$;
$\gamma_n \in \tilde E^n(S^n)$ the image of the multiplicative unit under the suspension isomorphism $\tilde E^0(S^0) \stackrel{\simeq}{\to}\tilde E^n(S^n)$.
(e.g. Kochmann 96, def. 4.3.4)
Let $E$ be a multiplicative cohomology theory and let $X$ be a manifold, possibly with boundary, of dimension $n$. An $E$-orientation of $X$ is a class in the $E$-generalized homology
with the property that for each point $x \in Int(X)$ in the interior, it maps to a generator of $E_\bullet(\ast)$ under the map
where the isomorphism is the excision isomorphism (def.) for the complement of a closed n-ball around $x$.
(e.g. Kochmann 96, p. 134)
$E$-orientations of manifolds (def. 2) are equivalent to $E$-orientations of their stable normal bundle (def. 1).
(e.g. Rudyak 98, chapter V, theorem 2.4) (also Kochmann 96, prop. 4.3.5, but maybe that proof needs an extra argument)
Recall that a (B,f)-structure $\mathcal{B}$ is a system of Serre fibrations $B_n \overset{f_n}{\longrightarrow} B O(n)$ over the classifying spaces for orthogonal structure equipped with maps
covering the canonical inclusions of classifying spaces. For instance for $G_n \to O(n)$ a compatible system of topological group homomorphisms, then the $(B,f)$-structure given by the classifying spaces $B G_n$ (possibly suitably resolved for the maps $B G_n \to B O(n)$ to become Serre fibrations) defines G-structure.
Given a $(B,f)$-structure, then there are the pullbacks $V^{\mathcal{B}}_n \coloneqq f_n^\ast (E O(n)\underset{O(n)}{\times}\mathbb{R}^n)$ of the universal vector bundles over $B O(n)$, which are the universal vector bundles equipped with $(B,f)$-structure
Finally recall that there are canonical morphisms (prop.)
Let $E$ be a multiplicative cohomology theory and let $\mathcal{B}$ be a multiplicative (B,f)-structure. Then a universal $E$-orientation for vector bundles with $\mathcal{B}$-structure is an $E$-orientation, according to def. 1, for each rank-$n$ universal vector bundle with $\mathcal{B}$-structure:
such that these are compatible in that
for all $n \in \mathbb{N}$ then
where
(with the first isomorphism is the suspension isomorphism of $E$ and the second exhibiting the homeomorphism of Thom spaces $Th(\mathbb{R} \oplus V)\simeq \Sigma Th(V)$ (prop.)) and where
is pullback along the canonical $\phi_n \colon \mathbb{R}\oplus V_n \to V_{n+1}$ (prop.).
for all $n_1, n_2 \in \mathbb{N}$ then
A universal $E$-orientation, in the sense of def. 3, for vector bundles with (B,f)-structure $\mathcal{B}$, is equivalently (the homotopy class of) a homomorphism of ring spectra
from the universal $\mathcal{B}$-Thom spectrum to a spectrum which via the Brown representability theorem represents the given generalized (Eilenberg-Steenrod) cohomology theory $E$ (and which we denote by the same symbol).
The Thom spectrum $M\mathcal{B}$ has a standard structure of a CW-spectrum. Let now $E$ denote a sequential Omega-spectrum representing the multiplicative cohomology theory of the same name. Since, in the standard model structure on topological sequential spectra, CW-spectra are cofibrant (prop.) and Omega-spectra are fibrant (thm.) we may represent all morphisms in the stable homotopy category (def.) by actual morphisms
of sequential spectra (due to this lemma).
Now by definition (def.) such a homomorphism is precissely a sequence of base-point preserving continuous functions
for $n \in \mathbb{N}$, such that they are compatible with the structure maps $\sigma_n$ and equivalently with their $(S^1 \wedge(-)\dashv Maps(S^1,-)_\ast)$-adjuncts $\tilde \sigma_n$, in that these diagrams commute:
for all $n \in \mathbb{N}$.
First of all this means (via the identification given by the Brown representability theorem, see this prop.) that the components $\xi_n$ are equivalently representatives of elements in the cohomology groups
(which we denote by the same symbol, for brevity).
Now by the definition of universal Thom spectra (def., def.), the structure map $\sigma_n^{M\mathcal{B}}$ is just the map $\phi_n \colon \mathbb{R}\oplus Th(V^{\mathcal{B}}_n)\to Th(V_{n+1}^{\mathcal{B}})$ from above.
Moreover, by the Brown representability theorem, the adjunct $\tilde \sigma_n^E \circ \xi_n$ (on the right) of $\sigma^E_n \circ S^1 \wedge \xi_n$ (on the left) is what represents (again by this prop.) the image of
under the suspension isomorphism. Hence the commutativity of the above squares is equivalently the first compatibility condition from def. 3: $\xi_n \simeq \phi_n^\ast \xi_{n+1}$ in $\tilde E^{n+1}(Th(\mathbb{R}\oplus V_n^{\mathcal{B}}))$
Next, $\xi$ being a homomorphism of ring spectra means equivalently (we should be modelling $M\mathcal{B}$ and $E$ as structured spectra (here.) to be more precise on this point, but the conclusion is the same) that for all $n_1, n_2\in \mathbb{N}$ then
This is equivalently the condition $\xi_{n_1} \cdot \xi_{n_2} \simeq \xi_{n_1 + n_2}$.
Finally, since $M\mathcal{B}$ is a ring spectrum, there is an essentially unique multiplicative homomorphism from the sphere spectrum
This is given by the component maps
that are induced by including the fiber of $V_{n}^{\mathcal{B}}$.
Accordingly the composite
has as components the restrictions $i^\ast \xi_n$ appearing in def. 1. At the same time, also $E$ is a ring spectrum, hence it also has an essentially unique multiplicative morphism $\mathbb{S} \to E$, which hence must agree with $i^\ast \xi$, up to homotopy. If we represent $E$ as a symmetric ring spectrum, then the canonical such has the required property: $e_0$ is the identity element in degree 0 (being a unit of an ordinary ring, by definition) and hence $e_n$ is necessarily its image under the suspension isomorphism, due to compatibility with the structure maps and using the above analysis.
Let $E$ be a E-∞ ring spectrum. Write $\mathbb{S}$ for the sphere spectrum.
For the following see also May,Sigurdsson: Parametrized Homotopy Theory (MO comment)
Write $R^\times$ or $GL_1(R)$ for the general linear group of the $E_\infty$-ring $R$: it is the subspace of the degree-0 space $\Omega^\infty R$ on those points that map to multiplicatively invertible elements in the ordinary ring $\pi_0(R)$.
Since $R$ is $E_\infty$, the space $GL_1(R)$ is itself an infinite loop space. Its one-fold delooping $B GL_1(R)$ is the classifying space for $GL_1(R)$-principal ∞-bundles (in Top): for $X \in Top$ and $\zeta : X \to B GL_1(R)$ a map, its homotopy fiber
is the $GL_1(R)$-principal $\infty$-bundle $P \to X$ classified by that map.
For $R = \mathbb{S}$ the sphere spectrum, we have that $B GL_1(\mathbb{S})$ is the classifying space for spherical fibrations.
There is a canonical morphism
from the classifying space of the orthogonal group to that of the infinity-group of units of the sphere spectrum, called the J-homomorphism. Postcomposition with this sends real vector bundles $V \to X$ to sphere bundles. This is what is modeled by the Thom space construction
which sends each fiber to its one-point compactification.
For $P \to X$ a $GL_1(R)$-principal ∞-bundle there is canonically the corresponding associated ∞-bundle with fiber $R$. Precisely, in the stable (∞,1)-category $Stab(Top)$ of spectra, regarded as the stabilization of the (∞,1)-topos Top
the associated bundle is the smash product over $\Sigma^\infty GL_1(R)$
This is the generalized Thom spectrum. For $R = K O$ the real K-theory spectrum this is given by the ordinary Thom space construction on a vector bundle $V \to X$.
An $E$-orientation of a vector bundle $V \to X$ is a trivialization of the $E$-module bundle $E \wedge S^V$, where we fiberwise form the smash product of $E$ with the Thom space of $V$.
For $f : R \to S$ a morphism of $E_\infty$-rings, and $\zeta : X \to B GL_1(R)$ the classifying map for an $R$-bundle, the corresponding associated $S$-bundle classified by the composite
is given by the smash product
This appears as (Hopkins, bottom of p. 6).
For $X \stackrel{\zeta}{\to} B GL_1(\mathbb{S})$ a sphere bundle, an $R$-orientation on $X^\zeta$ is a trivialization of the associated $R$-bundle $X^\zeta \wedge R$, hence a trivialization (null-homotopy) of the classifying morphism
where the second map comes from the unit of $E_\infty$-rings $\mathbb{S} \to R$ (the sphere spectrum is the initial object in $E_\infty$-rings).
Specifically, for $V : X \to B O$ a vector bundle, an $E$-orientation on it is a trivialization of the $R$-bundle associated to the associated Thom space sphere bundle, hence a trivialization of the morphism
This appears as (Hopkins, p.7).
A natural $R$-orientation of all vector bundles is therefore a trivialization of the morphism
Similarly, an $R$-orientation of all spinor bundles is a trivialization of
and an $R$-orientation of all string group-bundles a trivialization of
and so forth, through the Whitehead tower of $B O$.
Now, the Thom spectrum MO is the spherical fibration over $B O$ associated to the $O$-universal principal bundle. In generalization of the way that a trivialization of an ordinary $G$-principal bundle $P$ is given by a $G$-equivariant map $P \to G$, one finds that trivializations of the morphism
correspond to $E_\infty$-maps
from the Thom spectrum to $R$. Similarly trivialization of
corresponds to morphisms
and trivializations of
to morphisms
and so forth.
This is the way orientations in generalized cohomology often appear in the literature.
The construction of the string orientation of tmf, hence a morphism
is discussed in (Hopkins, last pages).
The relation (equivalence) between choices of Thom classes and trivializations of (∞,1)-line bundles is discussed e.g. in Ando-Hopkins-Rezk 10, section 3.3
Let $G$ be a topological group equipped with a homomorphism to the stable orthogonal group, and write $B G \to B O$ for the corresponding map of classifying spaces. Write $J \colon B O \longrightarrow B GL_1(\mathbb{S})$ for the J-homomorphism.
For $E$ an E-∞ ring, there is a canonical homomorphism $B GL_1(\mathbb{S}) \to B GL_1(E)$ between the deloopings of the ∞-groups of units. A trivialization of the total composite
is a universal $E$-orientation of G-structures. Under (∞,1)-colimit in $E Mod$ this induces a homomorphism of $E$-∞-modules
from the universal $G$-Thom spectrum to $E$.
If here $G \to GL_1(\mathbb{S})$ is the $\Omega^\infty$-component of a map of spectra then this $\sigma$ is a homomorphism of E-∞ rings and in this case there is a bijection between universal orientations and such $E_\infty$-ring homomorphisms (Ando-Hopkins-Rezk 10, prop. 2.11).
The latter, on passing to homotopy groups, are genera on manifolds with G-structure.
For $E$ a multiplicative weakly periodic complex orientable cohomology theory then $Spec E^0(B U\langle 6\rangle)$ is naturally equivalent to the space of cubical structures on the trivial line bundle over the formal group of $E$.
In particular, homotopy classes of maps of E-infinity ring spectra $MU\angle 6\rangle \to E$ from the Thom spectrum to $E$, and hence universal $MU\langle 6\rangle$-orientations (see there) of $E$ are in natural bijection with these cubical structures.
See at cubical structure for more details and references. This way for instance the string orientation of tmf has been constructed. See there for more.
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
An $E_\infty$ complex oriented cohomology theory $E$ is indeed equipped with a universal complex orientation given by an $E_\infty$-ring homomorphism $MU \to E$, see here.
Discussion in terms of Thom classes:
Frank Adams, part III, section 10 of Stable homotopy and generalised homology, 1974
Stanley Kochmann, section 4.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Yuli Rudyak, In Thom spectra, Orientability and Cobordism, Springer 1998 (pdf)
Jacob Lurie, lecture 5 of Chromatic Homotopy Theory, 2010 (pdf)
Manifold Atlas, Orientation of manifolds in generalized cohomology theories
A comprehensive account of the general abstract persepctive (with predecessors in Ando-Hopkins-Rezk 10) is in
Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)
Matthew Ando, Andrew Blumberg, David Gepner, section 6 of Twists of K-theory and TMF, in Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and $C^*$-algebras, Proceedings of Symposia in Pure Mathematics vol 81, American Mathematical Society (arXiv:1002.3004)
Lecture notes on this include
which are motivated towards constructing the string orientation of tmf, based on