nLab
fiber integration

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

In generalized cohomology by Umkehr maps

Along maps of manifolds

Here is the rough outline of the construction:

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 $e:E\hookrightarrow {ℝ}^n$ for some $n\in \mathbb{N}$. From this one obtains an embedding

$$(p,e):E\hookrightarrow B\times {ℝ}^n.$$(p,e) : E \hookrightarrow B \times \mathbb{R}^n \,.

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 $B\times {ℝ}^n$.

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

$$B\times {ℝ}^n\to B\times {ℝ}^n/(B\times {ℝ}^n-N_{(p,e)}(E))\simeq \mathrm{Th}(N_{(p,e)}(E))$$B \times \mathbb{R}^n \to B \times \mathbb{R}^n / (B \times \mathbb{R}^n - N_{(p,e)}(E)) \simeq Th(N_{(p,e)}(E))

factors through the one-point compactification $(B\times {ℝ}^n{)}^{*}$ of $B\times {ℝ}^n$. Since $(B\times {ℝ}^n{)}^{*}\cong {\Sigma }^n{B}_{+}$, 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

$$B\times {ℝ}^n\to {\Sigma }^n{B}_{+}\stackrel{\tau }{\to }\mathrm{Th}(N_{(p,e)}(E)),$$B \times \mathbb{R}^n \to \Sigma^n B_+ \stackrel{\tau}{\to} Th(N_{(p,e)}(E)) \,,

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

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

$$\tau :{\Sigma }^n{B}_{+}\stackrel{}{\to }\mathrm{Th}(N_{(p,e)}(E))$$\tau : \Sigma^n B_+ \stackrel{}{\to} Th(N_{(p,e)}(E))

is defined by

$$\tau :x\mapsto \{\begin{array}{cc}\varphi (x)& \mid x\in U\\ \mathrm{\infty }& \mid \mathrm{otherwise}\end{array}.$$\tau : x \mapsto \left\{ \array{ \phi(x) & | x \in U \\ \infty & | otherwise } \right. \,.

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

$${\int }_F:H^{•}(E)\to H^{•-\dim F}(B)$$\int_F : H^\bullet(E) \to H^{\bullet - dim F}(B)

of cohomologies

$$\begin{array}{c}{H}^{•}(E)\\ {\downarrow }^{{\simeq }_{\mathrm{Thom}}\to }\\ H^{•+n-\dim F}(D(N_{(p,e)}(E)),S(N_{(p,e)}(E)))\\ {\downarrow }^{\simeq }\\ {\tilde{H}}^{•+n-\dim F}(\mathrm{Th}(N_{(p,e)}(E)))& \stackrel{{\tau }^{*}}{\to }& {\tilde{H}}^{•+n-\dim F}({\Sigma }^n{B}_{+})\\ & & \downarrow {\simeq }_{\mathrm{suspension}}\\ & & H^{•-\dim F}(B)\end{array}.$$\array{ H^\bullet(E) \\ \downarrow^{\mathrlap{\simeq_{Thom}}{\to}} \\ H^{\bullet + n - dim F}(D(N_{(p,e)}(E)),S(N_{(p,e)}(E))) \\ \downarrow^{\mathrlap{\simeq}} \\ \tilde H^{\bullet + n - dim F}(Th(N_{(p,e)}(E))) & \stackrel{\tau^*}{\to} & \tilde H^{\bullet + n - dim F}(\Sigma^n B_+) \\ && \downarrow{\mathrlap{\simeq_{suspension}}} \\ && H^{\bullet - dim F}(B) } \,.

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 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).

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

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 $h:X\hookrightarrow Y$ be an embedding of compact smooth manifolds.

The push-forward constructed from this is supposed to be an element in KK-theory

$$h!:{\mathrm{KK}}_d(C(X),C(Y))$$h! \colon KK_d(C(X), C(Y))

in terms of which the push-forward on operator K-theory is induced by postcomposition:

$$h_{!}:K^{•}(X)\simeq {\mathrm{KK}}_{•}(ℂ,X)\stackrel{h!\circ (-)}{\to }{\mathrm{KK}}_{•+d}(ℂ,Y)\simeq {\mathrm{KK}}^{•+d}(Y),$$h_! \;\colon\; K^\bullet(X) \simeq KK_\bullet(\mathbb{C}, X) \stackrel{h!\circ (-)}{\to} KK_{\bullet+d}(\mathbb{C},Y) \simeq KK^{\bullet+d}(Y) \,,

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

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

$$h:X\hookrightarrow U\stackrel{j}{\hookrightarrow }Y$$h \;\colon\; X \hookrightarrow U \stackrel{j}{\hookrightarrow} Y

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

$$j!\in \mathrm{KK}(C_0(U),C(Y))$$j! \in KK(C_0(U), C(Y))

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

$$\psi :C(X)\stackrel{{\simeq }_{\mathrm{KK}}}{\to }{C}_0(U).$$\psi \colon C(X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,.

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

$$N_{Y}X≔h^{*}(TY)/TX$$N_Y X \coloneqq h^\ast(T Y)/ T X

has a spin^c structure. Write $S(N_{Y}X)$ for the associated spinor bundle.

Then there is an invertible element in KK-theory

$${\iota }^X!\in {\mathrm{KK}}_n(C(X),C_0(N_{Y}X))$$\iota^X! \in KK_n(C(X), C_0(N_Y X))

hence a KK-equivalence ${\iota }^X!:C(X)\stackrel{\simeq }{\to }{C}_0(N_{Y}X)$, where $C_0(-)$ denotes the algebra of functions vanishing at infinity.

This is defined as follows. Consider the pullback ${\pi }_n^{*}S(N_{Y}X)\to N_{Y}X$ of this spinor to the normal bundle itself along the projection ${\pi }_N:N_{Y}X\to X$. 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

$$\Phi :U_{h(X)}\hookrightarrow N_{Y}X.$$\Phi \colon U_{h(X)} \hookrightarrow N_Y X \,.

This induces a KK-equivalence

$$[\Phi ]:C_0(N_{Y}X)\stackrel{{\simeq }_{\mathrm{KK}}}{\to }{C}_0(U).$$[\Phi] \colon C_0(N_Y X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,.

Therefore the push-forward in operator K-theory along $f:X\hookrightarrow Y$ is given by postcomposing in KK-theory with

$$h!:C(X)\underset{{\simeq }_{\mathrm{KK}}}{\overset{i^X!}{\to }}{C}_0(N_{Y}X)\underset{{\simeq }_{\mathrm{KK}}}{\overset{\Phi }{\to }}{C}_0(U)\stackrel{j!}{\to }C(Y).$$h! \colon C(X) \underoverset{\simeq_{KK}}{i^X!}{\to} C_0(N_Y X) \underoverset{\simeq_{KK}}{\Phi}{\to} C_0(U) \stackrel{j!}{\to} C(Y) \,.

Along a proper submersion

(Connes-Skandalis 84, above prop. 2.9)

For $\pi :X\to Z$ 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 ${ℝ}^{2q}$ such as to yield an embedding

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

2. using that there is a KK-equivalence

   $${\iota }^Z!:C(Z)\stackrel{{\simeq }_{\mathrm{KK}}}{\to }{C}_0(Z\times {ℝ}^{2q}).$$\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

$$\pi !:C(X)\stackrel{h!}{\to }{C}_0(Z\times {ℝ}^{2}q)\underset{{\simeq }_{\mathrm{KK}}}{\overset{({\iota }^Z!{)}^{-1}}{\to }}C(Z).$$\pi! \colon C(X) \stackrel{h!}{\to} C_0(Z \times \mathbb{R}^{2}q) \underoverset{\simeq_{KK}}{(\iota^Z!)^{-1}}{\to} C(Z) \,.

(BMRS 07, example 3.4)

Along a smooth fibration of closed ${\mathrm{Spin}}^c$-manifolds

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

• closed smooth spin^c manifolds of even dimension

the push-forward element $\pi !\in \mathrm{KK}(C_0(X),C_0(Z))$ 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

$$T(X/Z)\hookrightarrow TX$$T(X/Z) \hookrightarrow T X

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 $TX\simeq T^{H}X\oplus T(X/Z)$ induces an affine connection ${\nabla }^{X/Z}$. This combined with the symbol map/Clifford multiplication of $T^{*}(X/Z)$ 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

$$(D_{X/Z},L^2(X/Z,S_{X/Z}))$$(D_{X/Z}, L^2(X/Z, S_{X/Z}))

which defines an element in KK-theory

$$\pi !\in \mathrm{KK}(C_0(X),C_0(Z)).$$\pi ! \in KK(C_0(X), C_0(Z)) \,.

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

$$f:X\stackrel{\mathrm{graph}(f)}{\to }X\times Y\stackrel{p_Y}{\to }Y.$$f \;\colon\; X \stackrel{graph(f)}{\to} X \times Y \stackrel{p_Y}{\to} Y \,.

Hence push-forward along such a general map is postcomposition in KK-theory with

$$f!≔p_Y!\circ \mathrm{graph}(f)!.$$f! \coloneqq p_Y !\circ graph(f)! \,.

(BMRS 07, example 3.5)

Examples

To the point

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

$${\int }_E:H^{•}(E)\to H^{•-\dim E}(*)$$\int_E:H^\bullet(E)\to H^{\bullet-dim 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

• geometric quantization by push-forward

The following terms all refer to essentially the same concept:

• fiber integration in generalized cohomology

• push-forward in generalized cohomology

• Umkehr map

• Gysin map

• Pontrjagin-Thom collapse map

References

General

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

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