|analytic integration||cohomological integration|
|measurable space||Poincaré duality|
|measure||orientation in generalized cohomology|
|volume form||(virtual) fundamental class|
|Riemann/Lebesgue integration of differential forms||push-forward in generalized cohomology/in differential cohomology|
More generally, for an -algebra spectrum, an -bundle is -orientable if the associated -bundle is trivializable. For more on this see (∞,1)-vector bundle.
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 .
is the -principal -bundle classified by that map.
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 to sphere bundles. This is what is modeled by the Thom space construction
which sends each fiber to its one-point compactification.
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
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).
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 is the sphere bundle 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.
The construction of the string orientation of tmf, hence a morphism
is discussed in (Hopkins, last pages).
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.
For spin orientation of vector bundle the construction is given by forming Clifford algebra bundles.
A comprehesive account is in
The general abstract story of -orientation of sphere fibrations is discussed in
Orientation of vector bundles in -cohomology is discussed for instance in