# nLab fiber integration

### Context

#### Integration theory

integration

analytic integrationcohomological integration
measureorientation in generalized cohomology
Riemann/Lebesgue integration, of differential formspush-forward in generalized cohomology/in differential cohomology

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

$\left(p,e\right):E↪B×{ℝ}^{n}\phantom{\rule{thinmathspace}{0ex}}.$(p,e) : E \hookrightarrow B \times \mathbb{R}^n \,.

Let ${N}_{\left(p,e\right)}\left(E\right)$ be the normal bundle of $E$ relative to this embedding. It is a rank $n-\mathrm{dim}F$ bundle over the image of $E$ in $B×{ℝ}^{n}$.

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

$B×{ℝ}^{n}\to B×{ℝ}^{n}/\left(B×{ℝ}^{n}-{N}_{\left(p,e\right)}\left(E\right)\right)\simeq \mathrm{Th}\left({N}_{\left(p,e\right)}\left(E\right)\right)$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 $\left(B×{ℝ}^{n}{\right)}^{*}$ of $B×{ℝ}^{n}$. Since $\left(B×{ℝ}^{n}{\right)}^{*}\cong {\Sigma }^{n}{B}_{+}$, 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

$B×{ℝ}^{n}\to {\Sigma }^{n}{B}_{+}\stackrel{\tau }{\to }\mathrm{Th}\left({N}_{\left(p,e\right)}\left(E\right)\right)\phantom{\rule{thinmathspace}{0ex}},$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 ${\Sigma }^{n}{B}_{+}\simeq B×{ℝ}^{n}\cup \left\{\infty \right\}$ and $\mathrm{Th}\left({N}_{\left(e,p\right)}\left(E\right)\right)={N}_{\left(e,p\right)}\cup \left\{\infty \right\}$, and for $U\subset {\Sigma }^{n}{B}_{+}$ a tubular neighbourhood of $E$ and $\varphi :U\to {N}_{\left(e,p\right)}\left(E\right)$ an isomorphism, the map

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

is defined by

$\tau :x↦\left\{\begin{array}{cc}\varphi \left(x\right)& \mid x\in U\\ \infty & \mid \mathrm{otherwise}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\mathrm{Th}\left({N}_{\left(p,e\right)}\left(E\right)\right)$ has an $H$-orientation, so that we have a Thom isomorphism. Then combined with the suspension isomorphism the pullback along $\tau$ produces a morphism

${\int }_{F}:{H}^{•}\left(E\right)\to {H}^{•-\mathrm{dim}F}\left(B\right)$\int_F : H^\bullet(E) \to H^{\bullet - dim F}(B)

of cohomologies

$\begin{array}{c}{H}^{•}\left(E\right)\\ {↓}^{{\simeq }_{\mathrm{Thom}}\to }\\ {H}^{•+n-\mathrm{dim}F}\left(D\left({N}_{\left(p,e\right)}\left(E\right)\right),S\left({N}_{\left(p,e\right)}\left(E\right)\right)\right)\\ {↓}^{\simeq }\\ {\stackrel{˜}{H}}^{•+n-\mathrm{dim}F}\left(\mathrm{Th}\left({N}_{\left(p,e\right)}\left(E\right)\right)\right)& \stackrel{{\tau }^{*}}{\to }& {\stackrel{˜}{H}}^{•+n-\mathrm{dim}F}\left({\Sigma }^{n}{B}_{+}\right)\\ & & ↓{\simeq }_{\mathrm{suspension}}\\ & & {H}^{•-\mathrm{dim}F}\left(B\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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.

(…)

See

## Examples

### To the point

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

${\int }_{E}:{H}^{•}\left(E\right)\to {H}^{•-\mathrm{dim}E}\left(*\right)$\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 $E$. …

## References

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

Revised on November 7, 2012 21:34:06 by Urs Schreiber (82.169.65.155)