fiber integration


Integration theory



Special and general types

Special notions


Extra structure





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

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:

  1. start with a map EBE \to B

  2. make EE bigger by passing to its Thom space Th(E)Th(E) such that we have a map the other way round BTh(E)B \to Th(E);

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

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

Now in detail.

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

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

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

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

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

B× nB× n/(B× nN (p,e)(E))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× n) *(B \times \mathbb{R}^n)^* of B× nB \times \mathbb{R}^n. Since (B× n) *Σ nB +(B \times \mathbb{R}^n)^*\cong \Sigma^n B_+, the nn-fold suspension of B +B_+ (or, equivalently, the smash product of BB with the nn-sphere: Σ nB +=S nB +\Sigma^n B_+= S^n \wedge B_+), we obtain a factorization

B× nΣ nB +τTh(N (p,e)(E)), B \times \mathbb{R}^n \to \Sigma^n B_+ \stackrel{\tau}{\to} Th(N_{(p,e)}(E)) \,,

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

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

τ:Σ nB +Th(N (p,e)(E)) \tau : \Sigma^n B_+ \stackrel{}{\to} Th(N_{(p,e)}(E))

is defined by

τ:x{ϕ(x) xU otherwise. \tau : x \mapsto \left\{ \array{ \phi(x) & | x \in U \\ \infty & | otherwise } \right. \,.

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

F:H (E)H dimF(B) \int_F : H^\bullet(E) \to H^{\bullet - dim F}(B)

of cohomologies

H (E) Thom H +ndimF(D(N (p,e)(E)),S(N (p,e)(E))) H˜ +ndimF(Th(N (p,e)(E))) τ * H˜ +ndimF(Σ nB +) suspension H dimF(B). \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 HH-cohomology along p:EBp : 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



D() Σ + L wheTop𝕊Mod D \coloneqq (-)^\vee\circ \Sigma^\infty_+ \coloneqq L_{whe} Top \to \mathbb{S}Mod

for the Spanier-Whitehead duality map which sends a topological space first to its suspension spectrum and then that to its dual object in the (∞,1)-category of spectra.

(ABG 11, def 10.3).


For XX a compact manifold, let X nX \to \mathbb{R}^n be an embedding and write S nX ν nS^n \to X^{\nu_n} for the classical Pontryagin-Thom collapse map for this situation, and write

𝕊X TX \mathbb{S} \to X^{-T X}

for the corresponding looping map from the sphere spectrum to the Thom spectrum of the negative tangent bundle of XX. Then Atiyah duality produces an equivalence

X TXDX X^{- T X} \simeq D X

which identifies the Thom spectrum with the dual object of Σ + X\Sigma^\infty_+ X in 𝕊Mod\mathbb{S} Mod and this constitutes a commuting diagram

X TX 𝕊 D(X*) DX \array{ && X^{- T X} \\ & \nearrow & \downarrow^{\mathrlap{\simeq}} \\ \mathbb{S} &\underset{D(X \to \ast)}{\to}& D X }

identifying the classical Pontryagin-Thom collapse map with the abstract dual morphism construction of prop. 1.

More generally, for WXW \hookrightarrow X an embedding of manifolds, then Atiyah duality identifies the Pontryagin-Thom collapse maps

𝕊X TXW TW \mathbb{S} \to X^{-T X} \to W^{- T W}

with the abstract dual morphisms

𝕊DXDW. \mathbb{S} \to D X \to D W \,.

(ABG 11, prop. 10.5).


Given now ECRing E \in CRing_\infty an E-∞ ring, then the dual morphism 𝕊DX\mathbb{S} \to D X induces under smash product a similar Pontryagin-Thom collapse map, but now not in sphere spectrum-(∞,1)-modules but in EE-(∞,1)-modules.

EDX 𝕊E. E \to D X \otimes_{\mathbb{S}} E \,.

The image of this under the EE-cohomology functor produces

[DX 𝕊E,E]E. [D X \otimes_{\mathbb{S}} E, E] \to E \,.

If now one has a Thom isomorphism (EE-orientation) [DX 𝕊E,E][X,E] [D X \otimes_{\mathbb{S}} E, E] \simeq [X,E] that identifies the cohomology of the dual object with the original cohomology, then together with produces the Umkehr map

[X,E][DX 𝕊E,E]E [X,E] \simeq [D X \otimes_{\mathbb{S}} E, E] \to E

that pushes the EE-cohomology of XX to the EE-cohomology of the point. Analogously if instead of the terminal map X*X \to \ast we start with a more general map XYX \to Y.

More generally a Thom isomorphism may not exists, but [DX 𝕊E,E][D X \otimes_{\mathbb{S}} E, E] may still be equivalent to a twisted cohomology-variant [X,E] χ[X,E]_{\chi} of [X,E][X,E], namely to [Γ X(χ),E][\Gamma_X(\chi),E], where χ:Π(X)ELineEMod\chi \colon \Pi(X) \to E Line \hookrightarrow E Mod is an (flat) EE-(∞,1)-module bundle on XX and and Γlim\Gamma \simeq \underset{\to}{\lim} is the (∞,1)-colimit (the generalized Thom spectrum construction). In this case the above yields a twisted Umkehr map.

(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).


twisted generalized cohomology theory is ∞-categorical semantics of linear homotopy type theory:

linear homotopy type theorygeneralized cohomology theoryquantum theory
linear type(module-)spectrum
multiplicative conjunctionsmash product of spectracomposite system
dependent linear typemodule spectrum bundle
Frobenius reciprocitysix operation yoga in Wirthmüller context
dual type (linear negation)Spanier-Whitehead duality
invertible typetwistprequantum line bundle
dependent sumgeneralized homology spectrumspace of quantum states (“bra”)
dual of dependent sumgeneralized cohomology spectrumspace of quantum states (“ket”)
linear implicationbivariant cohomologyquantum operators
exponential modalityFock space
dependent sum over finite homotopy type (of twist)suspension spectrum (Thom spectrum)
dualizable dependent sum over finite homotopy typeAtiyah duality between Thom spectrum and suspension spectrum
(twisted) self-dual typePoincaré dualityinner product
dependent sum coinciding with dependent productambidexterity, semiadditivity
dependent sum coinciding with dependent product up to invertible typeWirthmüller isomorphism
( ff *)(\sum_f \dashv f^\ast)-counitpushforward in generalized homology
(twisted-)self-duality-induced dagger of this counit(twisted-)Umkehr map/fiber integration
linear polynomial functorcorrespondencespace of trajectories
linear polynomial functor with linear implicationintegral kernel (pure motive)prequantized Lagrangian correspondence/action functional
composite of this linear implication with daggered-counit followed by unitintegral transformmotivic/cohomological path integral
traceEuler characteristicpartition function


In generalized differential cohomology


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

  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:XYh \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

h!: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)KK (,X)h!()KK +d(,Y)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)d = dim(X) - dim(Y).

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

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

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

j!KK(C 0(U),C(Y)) j! \in KK(C_0(U), C(Y))

induced directly from the C*-algebra homomorphism C 0(U)C(Y)C_0(U) \to C(Y) from the algebra of functions vanishing at infinity of UU to functions on YY, given by extending these functions by 0 to functions on YY. Or rather, it will be that element composed with the assumed KK-equivalence

ψ:C(X) KKC 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 YXh *(TY)/TX N_Y X \coloneqq h^\ast(T Y)/ T X

has a spin^c structure. Write S(N YX)S(N_Y X) for the associated spinor bundle.

Then there is an invertible element in KK-theory

ι X!KK n(C(X),C 0(N YX)) \iota^X! \in KK_n(C(X), C_0(N_Y X))

hence a KK-equivalence ι X!:C(X)C 0(N YX)\iota^X! \colon C(X) \stackrel{\simeq}{\to} C_0(N_Y X), where C 0()C_0(-) denotes the algebra of functions vanishing at infinity.

This is defined as follows. Consider the pullback π n *S(N YX)N YX\pi_n^\ast S(N_Y X) \to N_Y X of this spinor to the normal bundle itself along the projection π N:N YXX\pi_N \colon N_Y X \to X. Then

Moreover, a choice of a Riemannian metric on XX allows to find a diffeomorphism between the tubular neighbourhood U h(X)U_{h(X)} of h(X)h(X) and a neighbourhood of the zero-section of of the normal bundle

Φ:U h(X)N YX. \Phi \colon U_{h(X)} \hookrightarrow N_Y X \,.

This induces a KK-equivalence

[Φ]:C 0(N YX) KKC 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:XYf \colon X \hookrightarrow Y is given by postcomposing in KK-theory with

h!:C(X) KKi X!C 0(N YX) KKΦC 0(U)j!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 π:XZ\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 XX may be embedded into some 2q\mathbb{R}^{2q} such as to yield an embedding

    h:XZ× 2q h \colon X \to Z \times \mathbb{R}^{2 q}
  2. using that there is a KK-equivalence

    ι Z!:C(Z) KKC 0(Z× 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

π!:C(X)h!C 0(Z× 2q) KK(ι Z!) 1C(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 Spin cSpin^c-manifolds

Specifically, for π:XZ\pi \colon X \to Z a smooth fibration over a closed smooth manifold whose fibers X/ZX/Z are

the push-forward element π!KK(C 0(X),C 0(Z))\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

T(X/Z)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/Zg^{X/Z} on this bundle (hence a collection of Riemannian metric on the fibers X/ZX/Z smoothly varying along ZZ). Write S X/ZS_{X/Z} for the corresponding spinor bundle.

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

π!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:XYf \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 YY:

f:Xgraph(f)X×Yp YY. 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!graph(f)!. f! \coloneqq p_Y !\circ graph(f)! \,.

(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:QXi \colon Q \to X be a map of compact manifolds and let χ:XB 2U(1)\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

C i *χ(Q)i *C χ(X), C_{i^\ast \chi}(Q) \stackrel{i^\ast}{\longleftarrow} C_{\chi}(X) \,,

with notation as in this definition. By this proposition the dual morphism is of the form

C W 3(τ Q)i *χ(Q)i !C W 3(τ X)χ(X). C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{W_3(\tau_X)}{\chi}}(X) \,.

If we assume that XX has a spin^c structure then this is

C W 3(τ Q)i *χ(Q)i !C 1χ(X). C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{1}{\chi}}(X) \,.

Postcomposition with this map in KK-theory now yields a map from the W 3(τ Q)i *χ\frac{W_3(\tau_Q)}{i^\ast \chi}-twisted K-theory of QQ to the χ 1\chi^{-1}-twisted K-theory of XX:

i !:K +W 3(τ Q)i *χ(Q)K χ. i_! \colon K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) \to K_{\bullet -\chi} \,.

If we here think of i:QXi \colon Q \hookrightarrow X as being the inclusion of a D-brane worldvolume, then χ\chi would be the class of the background B-field and an element

[ξ]K +W 3(τ Q)i *χ(Q) [\xi] \in K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q)

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 ξ\xi given by a twisted line bundle in which case it exhibits a twisted spin^c structure on QQ.) Finally its push-forward

[i !ξ]K χ(X) [i_! \xi] \in K_{\bullet- \chi}(X)

is called the corresponding D-brane charge.


To the point

When BB is a point, one obtains integration aginst the fundamental class of EE,

E:H (E)H dimE(*) \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 Σ nB +=S n\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 EE and the Spanier-Whitehead dual of EE.

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 f!KK(C(X),C(Y))f! \in KK(C(X), C(Y)) for a KK-oriented map f:XYf \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:

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

Abstract formulation

The abstract formulation in stable homotopy theory via (infinity,1)-module bundles is sketched in section 9 of

and in section 10 of

This is reviewed and used also in

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

Revised on March 26, 2014 20:27:46 by Urs Schreiber (