Special and general types
Generally, for an E-∞ ring spectrum, and a sphere spectrum-bundle, an -orientation of is a trivialization of the associated -bundle.
Specifically, for the Thom space of a vector bundle , an -orientation of is an -orientation of .
More generally, for an -algebra spectrum, an -bundle is -orientable if the associated -bundle is trivializable. For more on this see (∞,1)-vector bundle.
The existence of an -orientation is necessary in order to have a notion of fiber integration in -cohomology.
Orientation of a vector bundle
Let be a multiplicative cohomology theory and let be a topological vector bundle of rank . Then an -orientation or -Thom class on is an element of degree
in the reduced -cohomology ring of the Thom space of , such that for every point its restriction along
(for the fiber of over ) is a generator, in that it is of the form
a unit in ;
the image of the multiplicative unit under the suspension isomorphism .
(e.g. Kochmann 96, def. 4.3.4)
Orientation of a manifold
Let be a multiplicative cohomology theory and let be a manifold, possibly with boundary, of dimension . An -orientation of is a class in the -generalized homology
with the property that for each point in the interior, it maps to a generator of under the map
where the isomorphism is the excision isomorphism (def.) for the complement of a closed n-ball around .
(e.g. Kochmann 96, p. 134)
-orientations of manifolds (def. 2) are equivalent to -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)
Universal orientation of vector bundles
Let be a multiplicative cohomology theory and let be a multiplicative (B,f)-structure. Then a universal -orientation for vector bundles with -structure is an -orientation, according to def. 1, for each rank- universal vector bundle with -structure:
such that these are compatible in that
for all then
(with the first isomorphism is the suspension isomorphism of and the second exhibiting the homeomorphism of Thom spaces (prop.)) and where
is pullback along the canonical (prop.).
for all then
The Thom spectrum has a standard structure of a CW-spectrum. Let now 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 , such that they are compatible with the structure maps and equivalently with their -adjuncts , in that these diagrams commute:
for all .
First of all this means (via the identification given by the Brown representability theorem, see this prop.) that the components 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 is just the map from above.
Moreover, by the Brown representability theorem, the adjunct (on the right) of (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: in
Next, being a homomorphism of ring spectra means equivalently (we should be modelling and as structured spectra (here.) to be more precise on this point, but the conclusion is the same) that for all then
This is equivalently the condition .
Finally, since 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 .
Accordingly the composite
has as components the restrictions appearing in def. 1. At the same time, also is a ring spectrum, hence it also has an essentially unique multiplicative morphism , which hence must agree with , up to homotopy. If we represent as a symmetric ring spectrum, then the canonical such has the required property: is the identity element in degree 0 (being a unit of an ordinary ring, by definition) and hence is necessarily its image under the suspension isomorphism, due to compatibility with the structure maps and using the above analysis.
Let be a E-∞ ring spectrum. Write for the sphere spectrum.
For the following see also May,Sigurdsson: Parametrized Homotopy Theory (MO comment)
Write or for the general linear group of the -ring : it is the subspace of the degree-0 space on those points that map to multiplicatively invertible elements in the ordinary ring .
Since is , the space is itself an infinite loop space. Its one-fold delooping is the classifying space for -principal ∞-bundles (in Top): for and a map, its homotopy fiber
is the -principal -bundle classified by that map.
For a -principal ∞-bundle there is canonically the corresponding associated ∞-bundle with fiber . Precisely, in the stable (∞,1)-category of spectra, regarded as the stabilization of the (∞,1)-topos Top
the associated bundle is the smash product over
This is the generalized Thom spectrum. For the real K-theory spectrum this is given by the ordinary Thom space construction on a vector bundle .
An -orientation of a vector bundle is a trivialization of the -module bundle , where we fiberwise form the smash product of with the Thom space of .
For a morphism of -rings, and the classifying map for an -bundle, the corresponding associated -bundle classified by the composite
is given by the smash product
This appears as (Hopkins, bottom of p. 6).
For a sphere bundle, an -orientation on is a trivialization of the associated -bundle , hence a trivialization (null-homotopy) of the classifying morphism
where the second map comes from the unit of -rings (the sphere spectrum is the initial object in -rings).
Specifically, for a vector bundle, an -orientation on it is a trivialization of the -bundle associated to the associated Thom space sphere bundle, hence a trivialization of the morphism
This appears as (Hopkins, p.7).
A natural -orientation of all vector bundles is therefore a trivialization of the morphism
Similarly, an -orientation of all spinor bundles is a trivialization of
and an -orientation of all string group-bundles a trivialization of
and so forth, through the Whitehead tower of .
Now, the Thom spectrum MO is the spherical fibration over associated to the -universal principal bundle. In generalization of the way that a trivialization of an ordinary -principal bundle is given by a -equivariant map , one finds that trivializations of the morphism
correspond to -maps
from the Thom spectrum to . Similarly trivialization of
corresponds to morphisms
and trivializations of
and so forth.
This is the way orientations in generalized cohomology often appear in the literature.
Relation between Thom classes and trivializations
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
Relation to genera
Let be a topological group equipped with a homomorphism to the stable orthogonal group, and write for the corresponding map of classifying spaces. Write for the J-homomorphism.
For an E-∞ ring, there is a canonical homomorphism between the deloopings of the ∞-groups of units. A trivialization of the total composite
is a universal -orientation of G-structures. Under (∞,1)-colimit in this induces a homomorphism of -∞-modules
from the universal -Thom spectrum to .
If here is the -component of a map of spectra then this is a homomorphism of E-∞ rings and in this case there is a bijection between universal orientations and such -ring homomorphisms (Ando-Hopkins-Rezk 10, prop. 2.11).
The latter, on passing to homotopy groups, are genera on manifolds with G-structure.
Relation to cubical structures
For a multiplicative weakly periodic complex orientable cohomology theory then is naturally equivalent to the space of cubical structures on the trivial line bundle over the formal group of .
In particular, homotopy classes of maps of E-infinity ring spectra from the Thom spectrum to , and hence universal -orientations (see there) of 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 complex oriented cohomology theory is indeed equipped with a universal complex orientation given by an -ring homomorphism , 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 -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