Contents

cohomology

# 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

### 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

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

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 $B \times \mathbb{R}^n$.

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 $B \times \mathbb{R}^n - N_{(p,e)}(E)$ to a point gives the Thom space of $N_{(p,e)}(E)$, and the quotient map

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

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

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

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

Explicitly, as sets we have $\Sigma^n B_+ \simeq B \times \mathbb{R}^n \cup \{\infty\}$ 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

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

is defined by

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

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

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

of cohomologies

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

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 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

###### Definition

Write

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

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.

###### Proposition

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

$\mathbb{S} \to X^{-T X}$

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

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

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

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

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

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

$\mathbb{S} \to X^{-T X} \to W^{- T W}$

with the abstract dual morphisms

$\mathbb{S} \to D X \to D W \,.$
###### Remark

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

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

The image of this under the $E$-cohomology functor produces

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

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

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

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

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

#### 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 conjecturally ∞-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
$(\sum_f \dashv f^\ast)$-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

(…)

See

### 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

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

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

$h! \colon KK_d(C(X), C(Y))$

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

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

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

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

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

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

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

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

$\psi \colon C(X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,.$

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

$N_Y X \coloneqq h^\ast(T Y)/ T X$

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

Then there is an invertible element in KK-theory

$\iota^X! \in KK_n(C(X), C_0(N_Y X))$

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

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

This induces a KK-equivalence

$[\Phi] \colon C_0(N_Y X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,.$

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

$h! \colon C(X) \underoverset{\simeq_{KK}}{i^X!}{\to} C_0(N_Y X) \underoverset{\simeq_{KK}}{\Phi}{\to} C_0(U) \stackrel{j!}{\to} C(Y) \,.$

#### Along a proper submersion

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

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

#### 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

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

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

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 $T X \simeq T^H X \oplus T(X/Z)$ 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

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

which defines an element in KK-theory

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

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

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 $Y$:

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

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

$f! \coloneqq p_Y !\circ graph(f)! \,.$

#### 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

$C_{i^\ast \chi}(Q) \stackrel{i^\ast}{\longleftarrow} C_{\chi}(X) \,,$

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

$C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{W_3(\tau_X)}{\chi}}(X) \,.$

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

$C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{1}{\chi}}(X) \,.$

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

$i_! \colon K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) \to K_{\bullet -\chi} \,.$

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

$[\xi] \in K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q)$

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

$[i_! \xi] \in K_{\bullet- \chi}(X)$

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 $E$,

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

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 $E$. …

The following terms all refer to essentially the same concept:

### General

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 $f! \in KK(C(X), C(Y))$ 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:

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