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.
Let be a E-∞ ring spectrum. Write for the sphere spectrum.
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 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.
Concretely for vector bundles
For the multiplicative cohomology theory corresponding to , and a vector bundle of rank , an -orientation of is an element in the cohomology of the Thom space of – a Thom class – with the property that its restriction along to any fiber of is
is a multiplicatively invertible element;
is the image of the multiplicative unit under the suspension isomorphism .
Multiplication with induces hence an isomorphism
This is called the Thom isomorphism.
The existence of an -orientation is necessary in order to have a notion of fiber integration in -cohomology.
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.
An complex oriented cohomology theory is indeed equipped with a universal complex orientation given by an -ring homomorphism , see here.
Ando-Hopkins-Rezk string orientation of
A comprehensive account of the general abstract persepctive (with predecessors in Ando-Hopkins-Rezk 10) is in
- 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 include
which are motivated towards constructing the string orientation of tmf, based on
Orientation of vector bundles in -cohomology is discussed for instance in