Contents

Idea

Fiber integration or push-forward is a process that sends generalized cohomology classes on a bundle $E \to B$ of manifolds to cohomology classes on the base $B$ 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 $k$-dimensional fibers reduces the degree of the cohomology class by $k$.

Composing pullback of cohomology classes with fiber integration yields the notion of transgression.

Definition

Fiberwise integration of ordinary differential forms

Consider a smooth submersion

of smooth manifolds, whose fibers have dimension $d$. The fiberwise integration of differential forms is a map

defined as follows. Given $\omega\in\Omega^n(T)$, the differential $(n-d)$-form $f_*(\omega)$ is constructed as follows.

The value of $f_*(\omega)$ at some collection of $(n-d)$ tangent vector fields $v_1$, …, $v_{n-d}$ is computed as follows.

First, lift each $v_i$ to a section of the quotient bundle $T(T)/T(f)$, where $T(f)$ is the relative tangent bundle?. Lift these sections to sections $u_i$ of $T(T)$ in some arbitrary way. Substitute the resulting vector fields on $T$ into the differential $n$-form $\omega$, obtaining a differential $d$-form $\psi$. Pull back $\psi$ to each of the fibers of the map $f\colon T\to B$, obtaining a $d$-form on each of these fibers, which does not depend on the choices of liftings $u_i$. Now integrate the resulting $d$-form over each of these fibers. This gives a number for each $b\in B$, which depends smoothly on $b$. This is the value of $f_*(\omega)$ on $v_1$, …, $v_{n-d}$.

See Greub, Halperin, and Vanstone, Volume I, Section VII.7.12.

In generalized cohomology via 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:

1. start with a map $E \to B$

2. make $B$ bigger without changing its homotopy type such that the map from $E$ becomes an embedding;

3. choose an orientation structure that makes the cohomology of $E$ equivalent to that of $Th(E)$ (the Thom isomorphism);

4. compose the Thom isomorphism with the pullback along $B \to Th(E)$ to get an “Umkehr” map from cohomology of $E$ to cohomology of $B$.

Now in detail.

Let $p : E \to B$ be a bundle of smooth compact manifolds with typical fiber $F$.

By the Whitney embedding theorem one can choose an embedding $e:E \hookrightarrow \mathbb{R}^n$ for some $n \in \mathbb{N}$. From this one obtains an embedding

Let $N_{(p,e)} (E)$ be the normal bundle of $E$ relative to this embedding. It is a rank $n- dim F$ bundle over the image of $E$ in $B \times \mathbb{R}^n$.

Fix a tubular neighbourhood of $E$ in $B \times \mathbb{R}^n$ and identify it with the total space of $N_{(p,e)}$. Then collapsing the whole complement $B \times \mathbb{R}^n \setminus N_{(p,e)}(E)$ to a point gives the Thom space of $N_{(p,e)}(E)$, and the quotient map

factors through the one-point compactification $(B \times \mathbb{R}^n)^*$ of $B \times \mathbb{R}^n$. Since $(B \times \mathbb{R}^n)^*\cong \Sigma^n B_+$ (see here), the $n$-fold suspension of $B_+$ (or, equivalently, the smash product of $B$ with the $n$-sphere: $\Sigma^n B_+= S^n \wedge B_+$), we obtain a factorization

where $\tau$ is called the Pontrjagin-Thom collapse map.

Explicitly, as sets we have $\Sigma^n B_+ \simeq B \times \mathbb{R}^n \cup \{\infty\}$ and $Th(N_{(e,p)}(E)) = N_{(e,p)} \cup \{\infty\}$, and for $U \subset \Sigma^n B_+$ a tubular neighbourhood of $E$ and $\phi : U \to N_{(e,p)}(E)$ an isomorphism, the map

is defined by

Now let $H$ be some multiplicative cohomology theory, and assume that the Thom space $Th(N_{(p,e)}(E))$ has an $H$-orientation, so that we have a Thom isomorphism. Then combined with the suspension isomorphism the pullback along $\tau$ produces a morphism

of cohomology rings:

This operation is independent of the choices involved. It is the fiber integration of $H$-cohomology along $p : E \to B$.

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 linear homotopy-type theory

We may formulate the above still a bit more abstractly in linear homotopy-type theory (following Homotopy-type semantics for quantization: see at indexed monoidal infinity-category the section on Fundamental classes and following.

(…)

In generalized differential cohomology

See

• fiber integration in differential cohomology

  • fiber integration in ordinary differential cohomology

  • fiber integration in differential K-theory

In KK-theory

We discuss fiber integration/push-forward/Gysin maps in operator K-theory, hence in KK-theory (Connes-Skandalis 85, BMRS 07, section 3). For more see at fiber integration in K-theory.

The following discusses KK-pushforward

1. Along an embedding

2. Along a submersion

3. Along a fibration of closed spin^c manifolds

4. Along a general K-oriented map

5. 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 $h \colon X \hookrightarrow Y$ 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:

where $d = dim(X) - dim(Y)$.

Now, if we could “thicken” $X$ a bit, namely to a tubular neighbourhood

of $h(X)$ in $Y$ without changing the K-theory of $X$, then the element in question will just be the KK-element

induced directly from the C*-algebra homomorphism $C_0(U) \to C(Y)$ from the algebra of functions vanishing at infinity of $U$ to functions on $Y$, given by extending these functions by 0 to functions on $Y$. 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 $S(N_Y X)$ for the associated spinor bundle.

Then there is an invertible element in KK-theory

hence a KK-equivalence $\iota^X! \colon C(X) \stackrel{\simeq}{\to} C_0(N_Y X)$, where $C_0(-)$ denotes the algebra of functions vanishing at infinity.

This is defined as follows. Consider the pullback $\pi_n^\ast S(N_Y X) \to N_Y X$ of this spinor to the normal bundle itself along the projection $\pi_N \colon N_Y X \to X$. Then…

Moreover, a choice of a Riemannian metric on $X$ allows to find a diffeomorphism between the tubular neighbourhood $U_{h(X)}$ of $h(X)$ 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 $f \colon X \hookrightarrow Y$ is given by postcomposing in KK-theory with

Along a proper submersion

(Connes-Skandalis 84, above prop. 2.9)

For $\pi \colon X \to Z$ 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

1. using that by the Whitney embedding theorem every compact $X$ may be embedded into some $\mathbb{R}^{2q}$ such as to yield an embedding

   $$h \colon X \to Z \times \mathbb{R}^{2 q}$$

2. using that there is a KK-equivalence

   $$\iota^Z! \colon C(Z) \stackrel{\simeq_{KK}}{\to} C_0(Z \times \mathbb{R}^{2q}) \,.$$

The resulting push-forward is then given by postcomposition in KK-theory with

(BMRS 07, example 3.4)

Along a smooth fibration of closed $Spin^c$-manifolds

Specifically, for $\pi \colon X \to Z$ a smooth fibration over a closed smooth manifold whose fibers $X/Z$ are

• closedsmooth spin^c manifolds of even dimension

the push-forward element $\pi! \in KK(C_0(X), C_0(Z))$ 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 $g^{X/Z}$ on this bundle (hence a collection of Riemannian metric on the fibers $X/Z$ smoothly varying along $Z$). Write $S_{X/Z}$ for the corresponding spinor bundle.

A choice of horizontal complenet $T X \simeq T^H X \oplus T(X/Z)$ induces an affine connection $\nabla^{X/Z}$. This combined with the symbol map/Clifford multiplication of $T^\ast (X/Z)$ on $S_{X/Z}$ induces a fiberwise spin^c Dirac operator, acting in each fiber on the Hilbert space $L^2(X/Z, S_{X/Z})$.

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 $f \colon X \to Y$ 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 $Y$:

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 $i \colon Q \to X$ be a map of compact manifolds and let $\chi \colon X \to B^2 U(1)$ 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 $X$ has a spin^c structure then this is

Postcomposition with this map in KK-theory now yields a map from the $\frac{W_3(\tau_Q)}{i^\ast \chi}$-twisted K-theory of $Q$ to the $\chi^{-1}$-twisted K-theory of $X$:

If we here think of $i \colon Q \hookrightarrow X$ as being the inclusion of a D-brane worldvolume, then $\chi$ 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 $\xi$ given by a twisted line bundle in which case it exhibits a twisted spin^c structure on $Q$.) Finally its push-forward

is called the corresponding D-brane charge.

Of cohesive differential form data

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.

Examples

To the point

When $B$ is a point, one obtains integration aginst the fundamental class of $E$,

taking values in the coefficients of the given cohomology theory. Note that in this case $\Sigma^n B_+=S^n$, 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 $E$ and the Spanier-Whitehead dual of $E$. …

Related concepts

• Grothendieck-Riemann-Roch theorem

• transfer context, sheaf with transfer, Becker-Gottlieb transfer

• geometric quantization by push-forward

• twisted Umkehr map

  • fiber integration in K-theory

References

General

• Frank Adams, pages 25-27 in Stable homotopy and generalized homology, Chicago Lectures in mathematics, 1974

• 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

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)

A quick summary can be found from slide 14 on in

• Robin Koytcheff, A homotopy-theoretic view of Bott-Taubes integrals (pdf slides)

More details are in

• Ralph Cohen, John Klein, Umkehr Maps (arXiv:0711.0540)

In noncommutative topology and KK-theory

Push-forward in twisted K-theory is discussed in

• Alan Carey, Bai-Ling Wang, Thom isomorphism and Push-forward map in twisted K-theory (arXiv:math/0507414)

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 $f! \in KK(C(X), C(Y))$ for a $K$-oriented map $f \colon X \to Y$ 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:

• Alain Connes, Georges Skandalis, The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. 20,

  no. 6, 1139–1183 (1984) (web)

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

• Paulo Carrillo Rouse, Bai-Ling Wang, Twisted longitudinal index theorem for foliations and wrong way functoriality (arXiv:1005.3842)

Dicussion for general C*-algebras is in section 3 of

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, Noncommutative correspondences, duality and D-branes in bivariant K-theory, Adv. Theor. Math. Phys.13:497-552,2009 (arXiv:0708.2648)

and specifically including also twisted K-theory again (and the relation to D-brane charge) in section 7 of

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, D-Branes, RR-Fields and Duality on Noncommutative Manifolds, Commun. Math. Phys. 277:643-706,2008 (arXiv:hep-th/0607020)

Abstract formulation

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 $C^*$-algebras, Proceedings of Symposia in Pure Mathematics vol 81, American Mathematical Society (arXiv:1002.3004)

and in section 10 of

• Matthew Ando, Andrew Blumberg, David Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map (arXiv:1112.2203)

This is reviewed and used also in

• Joost Nuiten, Cohomological quantization of boundary prequantum field theory, MSc thesis, Utrecht 2013

Formulation of this in linear homotopy-type theory is discussed in

• Urs Schreiber, Homotopy-type semantics for quantization