integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
Here is the rough outline of the construction via Pontryagin-Thom collapse maps.
The basic strategy is this:
start with a map $E \to B$
make $B$ bigger without changing its homotopy type such that the map from $E$ becomes an embedding;
choose an orientation structure that makes the cohomology of $E$ equivalent to that of $Th(E)$ (the Thom isomorphism);
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
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
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
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
is defined by
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
of cohomologies
This operation is independent of the choices involved. It is the fiber integration of $H$-cohomology along $p : E \to B$.
The above definition generalizes to one of push-forward in generalized cohomology on stacks over SmthMfd along representable morphisms of stacks.
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We discuss now a general abstract reformulation in terms of duality in stable homotopy theory and higher algebra of the above traditional constructions.
Write
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.
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
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
which identifies the Thom spectrum with the dual object of $\Sigma^\infty_+ X$ in $\mathbb{S} Mod$ and this constitutes a commuting diagram
identifying the classical Pontryagin-Thom collapse map with the abstract dual morphism construction of prop. 1.
More generally, for $W \hookrightarrow X$ an embedding of manifolds, then Atiyah duality identifies the Pontryagin-Thom collapse maps
with the abstract dual morphisms
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.
The image of this under the $E$-cohomology functor produces
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
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.
We may formulate the above still a bit more abstractly in linear homotopy-type theory (following Homotopy-type semantics for quantization).
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twisted generalized cohomology theory is ∞-categorical semantics of linear homotopy type theory:
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See
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).
(Connes-Skandalis 84, above prop. 2.8)
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
in terms of which the push-forward on operator K-theory is induced by postcomposition:
where $d = dim(X) - dim(Y)$.
Now, if we could “thicken” $X$ a bit, namely to a tubular neighbourhood
of $h(X)$ in $Y$ without changing the K-theory of $X$, then the element in question will just be the KK-element
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
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
has a spin^c structure. Write $S(N_Y X)$ for the associated spinor bundle.
Then there is an invertible element in KK-theory
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
This induces a KK-equivalence
Therefore the push-forward in operator K-theory along $f \colon X \hookrightarrow Y$ is given by postcomposing in KK-theory with
(Connes-Skandalis 84, above prop. 2.9)
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
using that by the Whitney embedding theorem every compact $X$ may be embedded into some $\mathbb{R}^{2q}$ such as to yield an embedding
using that there is a KK-equivalence
The resulting push-forward is then given by postcomposition in KK-theory with
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
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
which defines an element in KK-theory
Postcompositon with this is the push-forward map in K/KK-theory, equivalently the index map of the collection of Dirac operators.
(Connes-Skandalis 84, def. 2.1)
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 $Y$:
Hence push-forward along such a general map is postcomposition in KK-theory with
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
with notation as in this definition. By this proposition the dual morphism is of the form
If we assume that $X$ has a spin^c structure then this is
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 $X$:
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
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
is called the corresponding D-brane charge.
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.
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 $E$. …
The following terms all refer to essentially the same concept:
Frank Adams, pages 25-27 in Stable homotopy and generalized homology, Chicago Lectures in mathematics, 1974
Stanley Kochmann, section 4.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
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
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
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:
(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
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