|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|
Fiber integration or push-forward is a process that sends generalized cohomology classes on a bundle of manifolds to cohomology classes on the base of the bundle, by evaluating them on each fiber in some sense.
This sense is such that if the cohomology in question is de Rham cohomology then fiber integration is ordinary integration of differential forms over the fibers. Generally, the fiber integration over a bundle of -dimensional fibers reduces the degree of the cohomology class by .
Composing pullback of cohomology classes with fiber integration yields the notion of transgression.
Here is the rough outline of the construction via Pontryagin-Thom collapse maps.
The basic strategy is this:
start with a map
make bigger without changing its homotopy type such that the map from becomes an embedding;
compose the Thom isomorphism with the pullback along to get an “Umkehr” map from cohomology of to cohomology of .
Now in detail.
By the Whitney embedding theorem one can choose an embedding for some . From this one obtains an embedding
Let be the normal bundle of relative to this embedding. It is a rank bundle over the image of in .
where is called the Pontrjagin-Thom collapse map.
Explicitly, as sets we have and , and for a tubular neighbourhood of and an isomorphism, the map
is defined by
Now let be some multiplicative cohomology theory, and assume that the Thom space has an -orientation, so that we have a Thom isomorphism. Then combined with the suspension isomorphism the pullback along produces a morphism
This operation is independent of the choices involved. It is the fiber integration of -cohomology along .
with the abstract dual morphisms
The image of this under the -cohomology functor produces
that pushes the -cohomology of to the -cohomology of the point. Analogously if instead of the terminal map we start with a more general map .
More generally a Thom isomorphism may not exists, but may still be equivalent to a twisted cohomology-variant of , namely to , where is an (flat) -(∞,1)-module bundle on and and is the (∞,1)-colimit (the generalized Thom spectrum construction). In this case the above yields a twisted Umkehr map.
|linear homotopy type theory||generalized cohomology theory||quantum theory|
|multiplicative conjunction||smash product of spectra||composite system|
|dependent linear type||module spectrum bundle|
|Frobenius reciprocity||six operation yoga in Wirthmüller context|
|dual type (linear negation)||Spanier-Whitehead duality|
|invertible type||twist||prequantum line bundle|
|dependent sum||generalized homology spectrum||space of quantum states (“bra”)|
|dual of dependent sum||generalized cohomology spectrum||space of quantum states (“ket”)|
|linear implication||bivariant cohomology||quantum operators|
|exponential modality||Fock space|
|dependent sum over finite homotopy type (of twist)||suspension spectrum (Thom spectrum)|
|dualizable dependent sum over finite homotopy type||Atiyah duality between Thom spectrum and suspension spectrum|
|(twisted) self-dual type||Poincaré duality||inner product|
|dependent sum coinciding with dependent product||ambidexterity, semiadditivity|
|dependent sum coinciding with dependent product up to invertible type||Wirthmüller isomorphism|
|-counit||pushforward in generalized homology|
|(twisted-)self-duality-induced dagger of this counit||(twisted-)Umkehr map/fiber integration|
|linear polynomial functor||correspondence||space of trajectories|
|linear polynomial functor with linear implication||integral kernel (pure motive)||prequantized Lagrangian correspondence/action functional|
|composite of this linear implication with daggered-counit followed by unit||integral transform||motivic/cohomological path integral|
|trace||Euler characteristic||partition function|
The following discusses KK-pushforward
The construction goes back to (Connes 82), where it is given over smooth manifolds. Then (Connes-Skandalis 84, Hilsum-Skandalis 87) generalize this to maps between foliations by KK-elements betwen the groupoid convolution algebras of the coresponding holonomy groupoids and (Rouse-Wang 10) further generalize to the case where a circle 2-bundle twist is present over these foliations. A purely algebraic generalization to (K-oriented) maps between otherwise arbitrary noncommutative spaces/C*-algebras is in (BMRS 07).
The push-forward constructed from this is supposed to be an element in KK-theory
in terms of which the push-forward on operator K-theory is induced by postcomposition:
Now, if we could “thicken” a bit, namely to a tubular neighbourhood
of in without changing the K-theory of , then the element in question will just be the KK-element
induced directly from the C*-algebra homomorphism from the algebra of functions vanishing at infinity of to functions on , given by extending these functions by 0 to functions on . Or rather, it will be that element composed with the assumed KK-equivalence
The bulk of the technical work in constructing the push-forward is in constructing this equivalence. (BMRS 07, example 3.3)
In order for it to exist at all, assume that the normal bundle
Then there is an invertible element in KK-theory
This is defined as follows. Consider the pullback of this spinor to the normal bundle itself along the projection . Then…
This induces a KK-equivalence
Therefore the push-forward in operator K-theory along is given by postcomposing in KK-theory with
using that by the Whitney embedding theorem every compact may be embedded into some such as to yield an embedding
using that there is a KK-equivalence
The resulting push-forward is then given by postcomposition in KK-theory with
Specifically, for a smooth fibration over a closed smooth manifold whose fibers are
the push-forward element is given by the Fredholm-Hilbert module obatined from the fiberwise spin^c Dirac operator acting on the fiberwise spinors. (Connes-Skandalis 84, proof of lemma 4.7, BMRS 07, example 3.9).
In detail, write
for the sub-bundle of the total tangent bundle on the vertical vectors and choose a Riemannian metric on this bundle (hence a collection of Riemannian metric on the fibers smoothly varying along ). Write for the corresponding spinor bundle.
A choice of horizontal complenet induces an affine connection . This combined with the symbol map/Clifford multiplication of on induces a fiberwise spin^c Dirac operator, acting in each fiber on the Hilbert space .
which defines an element in KK-theory
Postcompositon with this is the push-forward map in K/KK-theory, equivalently the index map of the collection of Dirac operators.
Hence push-forward along such a general map is postcomposition in KK-theory with
If we assume that has a spin^c structure then this is
is called (the K-class of) a Chan-Paton gauge field on the D-brane satisfying the Freed-Witten-Kapustin anomaly cancellation mechanism. (The orginal Freed-Witten anomaly cancellation assumes given by a twisted line bundle in which case it exhibits a twisted spin^c structure on .) Finally its push-forward
is called the corresponding D-brane charge.
In differential cohomology realized in cohesive homotopy type theory there is a canonical fiber integration map for the curvature coefficients of a given diffential cohomology theory. See at integration of differential forms – In cohesive homotopy-type theory.
When is a point, one obtains integration aginst the fundamental class of ,
taking values in the coefficients of the given cohomology theory. Note that in this case , and this hints to a relationship between the Thom-Pontryagin construction and Spanier-Whitehead duality. And indeed Atiyah duality gives a homotopy equivalence between the Thom spectrum of the stable normal bundle of and the Spanier-Whitehead dual of . …
The following terms all refer to essentially the same concept:
Stanley Kochmann, section 4.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Fiber integration of differential forms is discussed in section VII of volume I of
A quick summary can be found from slide 14 on in
More details are in
Push-forward in twisted K-theory is discussed in
and section 10 of (ABG, 10)
The definition of the element for a -oriented map between smooth manifolds goes back to section 11 in
The functoriality of this construction is demonstrated in section 2 of the following article, which moreover generalizes the construction to maps between foliations hence to KK-elements between groupoid convolution algebras of holonomy groupoids:
(the article that introuced Hilsum-Skandalis morphisms).
This is further generalized to circle 2-bundle-twisted convolution algebras of foliations in
Dicussion for general C*-algebras is in section 3 of
and in section 10 of
This is reviewed and used also in
Formulation of this in linear homotopy-type theory is discussed in