Special and general types
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.
In generalized cohomology by Pontryagin-Thom collapse maps
Along maps of manifolds
Here is the rough outline of the construction via Pontryagin-Thom collapse maps.
The basic strategy is this:
start with a map
make bigger by passing to its Thom space such that we have a map the other way round ;
choose an orientation structure that makes the cohomology of equivalent to that of (the Thom isomorphism);
compose the Thom isomorphism with the pullback along to get an “Umkehr” map from cohomology of to cohomology of .
Now in detail.
Let be a bundle of smooth compact manifolds with typical fiber .
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 .
Fix a tubular neighbourhood of in and identify it with the total space of . Then collapsing the whole to a point gives the Thom space of , and the quotient map
factors through the one-point compactification of . Since , the -fold suspension of (or, equivalently, the smash product of with the -sphere: ), we obtain a factorization
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 .
Along representable morphisms of stacks
The above definition generalizes to one of push-forward in generalized cohomology on stacks over SmthMfd along representable morphisms of stacks.
In generalized cohomology by Umkehr maps via abstract duality
We discuss now a general abstract reformulation in terms of duality in stable homotopy theory and higher algebra of the above traditional constructions.
Abstract duality and Atiyah-Milnor-Spanier duality + Pontryagin-Thom collapse
(ABG 11, def 10.3).
(ABG 11, prop. 10.5).
(ABG 10, 9.1)
In generalized differential cohomology
We discuss fiber integration/push-forward/Gysin maps in operator K-theory, hence in KK-theory (Connes-Skandalis 85, BMRS 07, section 3).
The following discusses KK-pushforward
Along an embedding
Along a submersion
Along a fibration of closed spin^c manifolds
Along a general K-oriented map
In twisted K-theory
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).
Along an embedding
(Connes-Skandalis 84, above prop. 2.8)
Let be an embedding of compact smooth manifolds.
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
has a spin^c structure. Write for the associated spinor bundle.
Then there is an invertible element in KK-theory
hence a KK-equivalence , where denotes the algebra of functions vanishing at infinity.
This is defined as follows. Consider the pullback of this spinor to the normal bundle itself along the projection . Then…
Moreover, a choice of a Riemannian metric on allows to find a diffeomorphism between the tubular neighbourhood of and a neighbourhood of the zero-section of of the normal bundle
This induces a KK-equivalence
Therefore the push-forward in operator K-theory along is given by postcomposing in KK-theory with
Along a proper submersion
(Connes-Skandalis 84, above prop. 2.9)
For a K-oriented proper submersion of compact smooth manifolds, the push-forward map along it is reduced to the above case of an embedding by
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
(BMRS 07, example 3.4)
Along a smooth fibration of closed -manifolds
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 .
This yields a Fredholm-Hilbert bimodule
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.
Along a general K-oriented map
(Connes-Skandalis 84, def. 2.1)
Now for an arbitray K-oriented smooth proper map, we may reduce push-forward along it to the above two cases by factoring it through its graph map, followed by projection to :
Hence push-forward along such a general map is postcomposition in KK-theory with
(BMRS 07, example 3.5)
In twisted K-theory
We discuss push forward in K-theory more generally by Poincaré duality C*-algebras hence dual objects in KK-theory.
Let be a map of compact manifolds and let modulate a circle 2-bundle regarded as a twist for K-theory. Then forming twisted groupoid convolution algebras yields a KK-theory morphism of the form
with notation as in this definition. By this proposition the dual morphism is of the form
If we assume that has a spin^c structure then this is
Postcomposition with this map in KK-theory now yields a map from the -twisted K-theory of to the -twisted K-theory of :
If we here think of as being the inclusion of a D-brane worldvolume, then would be the class of the background B-field and an element
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.
To the point
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:
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
In noncommutative topology and KK-theory
Push-forward in twisted K-theory is discussed in
and section 10 of (ABG, 10)
Discussion of fiber integration Gysin maps/Umkehr maps in noncommutative topology/KK-theory as above is in the following references.
The definition of the element for a -oriented map between smooth manifolds goes back to section 11 in
- Alain Connes, A survey of foliations and operator algebras, Proceedings of the A.M.S., 38, 521-628 (1982) (pdf)
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:
More on this is in
- Michel Hilsum, Georges Skandalis, Morphismes K-orienté d’espace de feuille et fonctoralité en théorie de Kasparov, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 20 no. 3 (1987), p. 325-390 (numdam)
(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 specifically including also twisted K-theory again (and the relation to D-brane charge) in section 7 of
The abstract formulation in stable homotopy theory via (infinity,1)-module bundles is sketched in section 9 of
- Matthew Ando, Andrew Blumberg, David Gepner, 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)
and in section 10 of