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 $e:E\hookrightarrow {ℝ}^n$ 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 $B\times {ℝ}^n$.

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

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

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

is defined by

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

of cohomologies

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

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 $E$. …

References

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)

…