\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{fiber integration}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory}

[[!include integration theory - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_generalized_cohomology_via_pontryaginthom_collapse_maps}{In generalized cohomology via Pontryagin-Thom collapse maps}\dotfill \pageref*{in_generalized_cohomology_via_pontryaginthom_collapse_maps} \linebreak
\noindent\hyperlink{along_maps_of_manifolds}{Along maps of manifolds}\dotfill \pageref*{along_maps_of_manifolds} \linebreak
\noindent\hyperlink{along_representable_morphisms_of_stacks}{Along representable morphisms of stacks}\dotfill \pageref*{along_representable_morphisms_of_stacks} \linebreak
\noindent\hyperlink{in_generalized_cohomology_by_umkehr_maps_via_abstract_duality}{In generalized cohomology by Umkehr maps via abstract duality}\dotfill \pageref*{in_generalized_cohomology_by_umkehr_maps_via_abstract_duality} \linebreak
\noindent\hyperlink{abstract_duality_and_atiyahmilnorspanier_duality__pontryaginthom_collapse}{Abstract duality and Atiyah-Milnor-Spanier duality + Pontryagin-Thom collapse}\dotfill \pageref*{abstract_duality_and_atiyahmilnorspanier_duality__pontryaginthom_collapse} \linebreak
\noindent\hyperlink{in_linear_homotopytype_theory}{In linear homotopy-type theory}\dotfill \pageref*{in_linear_homotopytype_theory} \linebreak
\noindent\hyperlink{in_generalized_differential_cohomology}{In generalized differential cohomology}\dotfill \pageref*{in_generalized_differential_cohomology} \linebreak
\noindent\hyperlink{InKKTheory}{In KK-theory}\dotfill \pageref*{InKKTheory} \linebreak
\noindent\hyperlink{KKPushForwardAlongEmbedding}{Along an embedding}\dotfill \pageref*{KKPushForwardAlongEmbedding} \linebreak
\noindent\hyperlink{KKPushforwardAlongSubmersion}{Along a proper submersion}\dotfill \pageref*{KKPushforwardAlongSubmersion} \linebreak
\noindent\hyperlink{AlongAFibrationOfClosedSpinCManifolds}{Along a smooth fibration of closed $Spin^c$-manifolds}\dotfill \pageref*{AlongAFibrationOfClosedSpinCManifolds} \linebreak
\noindent\hyperlink{#KKPushforwardAlongGeneralMap}{Along a general K-oriented map}\dotfill \pageref*{#KKPushforwardAlongGeneralMap} \linebreak
\noindent\hyperlink{KKPushforwardAlongGeneralMap}{In twisted K-theory}\dotfill \pageref*{KKPushforwardAlongGeneralMap} \linebreak
\noindent\hyperlink{of_cohesive_differential_form_data}{Of cohesive differential form data}\dotfill \pageref*{of_cohesive_differential_form_data} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{to_the_point}{To the point}\dotfill \pageref*{to_the_point} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesInKKTheory}{In noncommutative topology and KK-theory}\dotfill \pageref*{ReferencesInKKTheory} \linebreak
\noindent\hyperlink{abstract_formulation}{Abstract formulation}\dotfill \pageref*{abstract_formulation} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\emph{Fiber integration} or \emph{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 \emph{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 form]]s 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]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_generalized_cohomology_via_pontryaginthom_collapse_maps}{}\subsubsection*{{In generalized cohomology via Pontryagin-Thom collapse maps}}\label{in_generalized_cohomology_via_pontryaginthom_collapse_maps}

\hypertarget{along_maps_of_manifolds}{}\paragraph*{{Along maps of manifolds}}\label{along_maps_of_manifolds}

Here is the rough outline of the construction via [[Pontryagin-Thom collapse maps]].

The basic strategy is this:

\begin{enumerate}%
\item start with a map $E \to B$


\item make $B$ bigger without changing its homotopy type such that the map from $E$ becomes an [[embedding]];


\item choose an [[orientation in generalized cohomology|orientation]] structure that makes the cohomology of $E$ equivalent to that of $Th(E)$ (the [[Thom isomorphism]]);


\item compose the Thom isomorphism with the pullback along $B \to Th(E)$ to get an ``Umkehr'' map from cohomology of $E$ to cohomology of $B$.



\end{enumerate}
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

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

\begin{displaymath}
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))
\end{displaymath}
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

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

\begin{displaymath}
\tau : \Sigma^n B_+ \stackrel{}{\to} Th(N_{(p,e)}(E))
\end{displaymath}
is defined by

\begin{displaymath}
\tau : x \mapsto 
  \left\{
    \itexarray{
      \phi(x) & | x \in U
      \\
      \infty & | otherwise
    }
  \right.
  \,.
\end{displaymath}
Now let $H$ be some [[multiplicative cohomology theory]], and assume that the Thom space $Th(N_{(p,e)}(E))$ has an $H$-[[orientation in generalized cohomology|orientation]], so that we have a [[Thom isomorphism]]. Then combined with the [[suspension isomorphism]] the pullback along $\tau$ produces a morphism

\begin{displaymath}
\int_F : H^\bullet(E) \to H^{\bullet - dim F}(B)
\end{displaymath}
of cohomologies

\begin{displaymath}
\itexarray{
   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)
   }
   \,.
\end{displaymath}
This operation is independent of the choices involved. It is the \textbf{fiber integration} of $H$-cohomology along $p : E \to B$.

\hypertarget{along_representable_morphisms_of_stacks}{}\paragraph*{{Along representable morphisms of stacks}}\label{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]].

(\ldots{})

\hypertarget{in_generalized_cohomology_by_umkehr_maps_via_abstract_duality}{}\subsubsection*{{In generalized cohomology by Umkehr maps via abstract duality}}\label{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.

\hypertarget{abstract_duality_and_atiyahmilnorspanier_duality__pontryaginthom_collapse}{}\paragraph*{{Abstract duality and Atiyah-Milnor-Spanier duality + Pontryagin-Thom collapse}}\label{abstract_duality_and_atiyahmilnorspanier_duality__pontryaginthom_collapse}

\begin{defn}
\label{SpanierDualityOperation}\hypertarget{SpanierDualityOperation}{}
Write

\begin{displaymath}
D \coloneqq (-)^\vee\circ \Sigma^\infty_+ \coloneqq L_{whe} Top \to \mathbb{S}Mod
\end{displaymath}
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]].

\end{defn}
(\hyperlink{ABG11}{ABG 11, def 10.3}).

\begin{prop}
\label{}\hypertarget{}{}
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

\begin{displaymath}
\mathbb{S} \to X^{-T X}
\end{displaymath}
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]]

\begin{displaymath}
X^{- T X} \simeq D X
\end{displaymath}
which identifies the [[Thom spectrum]] with the [[dual object]] of $\Sigma^\infty_+ X$ in $\mathbb{S} Mod$ and this constitutes a [[commuting diagram]]

\begin{displaymath}
\itexarray{
    && X^{- T X}
    \\
    & \nearrow & \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbb{S}
    &\underset{D(X \to \ast)}{\to}&
    D X
  }
\end{displaymath}
identifying the classical [[Pontryagin-Thom collapse map]] with the abstract [[dual morphism]] construction of prop. \ref{SpanierDualityOperation}.

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

\begin{displaymath}
\mathbb{S} \to X^{-T X} \to W^{- T W}
\end{displaymath}
with the abstract [[dual morphisms]]

\begin{displaymath}
\mathbb{S} \to D X \to D W
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{ABG11}{ABG 11, prop. 10.5}).

\begin{remark}
\label{}\hypertarget{}{}
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]].

\begin{displaymath}
E \to D X \otimes_{\mathbb{S}} E
  \,.
\end{displaymath}
The image of this under the $E$-[[generalized cohomology theory|cohomology]] functor produces

\begin{displaymath}
[D X \otimes_{\mathbb{S}} E, E] \to E
  \,.
\end{displaymath}
If now one has a [[Thom isomorphism]] ($E$-[[orientation in generalized cohomology|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 \textbf{Umkehr map}

\begin{displaymath}
[X,E] \simeq [D X \otimes_{\mathbb{S}} E, E] \to E
\end{displaymath}
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 (∞,1)-bundle|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 \emph{[[twisted Umkehr map]]}.

\end{remark}
(\hyperlink{ABG10}{ABG 10, 9.1})

\hypertarget{in_linear_homotopytype_theory}{}\paragraph*{{In linear homotopy-type theory}}\label{in_linear_homotopytype_theory}

We may formulate the above still a bit more abstractly in [[linear homotopy-type theory]] (following \emph{[[schreiber:Homotopy-type semantics for quantization]]}).

(\ldots{})

[[!include twisted generalized cohomology in linear homotopy type theory -- table]]

(\ldots{})

\hypertarget{in_generalized_differential_cohomology}{}\subsubsection*{{In generalized differential cohomology}}\label{in_generalized_differential_cohomology}

See

\begin{itemize}%
\item [[fiber integration in differential cohomology]]

\begin{itemize}%
\item [[fiber integration in ordinary differential cohomology]]


\item [[fiber integration in differential K-theory]]



\end{itemize}


\end{itemize}
\hypertarget{InKKTheory}{}\subsubsection*{{In KK-theory}}\label{InKKTheory}

We discuss fiber integration/push-forward/[[Gysin maps]] in [[operator K-theory]], hence in [[KK-theory]] (\hyperlink{ConnesSkandalis84}{Connes-Skandalis 85}, \hyperlink{BMRS07}{BMRS 07, section 3}). For more see at \emph{[[fiber integration in K-theory]].}

The following discusses KK-pushforward

\begin{enumerate}%
\item \emph{\hyperlink{KKPushForwardAlongEmbedding}{Along an embedding}}


\item \emph{\hyperlink{KKPushforwardAlongSubmersion}{Along a submersion}}


\item \emph{\hyperlink{AlongAFibrationOfClosedSpinCManifolds}{Along a fibration of closed spin{\tt \symbol{94}}c manifolds}}


\item \emph{\hyperlink{KKPushforwardAlongGeneralMap}{Along a general K-oriented map}}


\item \emph{\hyperlink{KKPushforwardAlongGeneralMap}{In twisted K-theory}}



\end{enumerate}
The construction goes back to (\hyperlink{Connes82}{Connes 82}), where it is given over smooth manifolds. Then (\hyperlink{ConnesSkandalis84}{Connes-Skandalis 84}, \hyperlink{HilsumSkandalis87}{Hilsum-Skandalis 87}) generalize this to maps between [[foliations]] by KK-elements betwen the [[groupoid convolution algebras]] of the coresponding [[holonomy groupoids]] and (\hyperlink{RouseWang10}{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 topology|noncommutative spaces]]/[[C\emph{-algebras]] is in (\hyperlink{BMRS07}{BMRS 07}).}

\hypertarget{KKPushForwardAlongEmbedding}{}\paragraph*{{Along an embedding}}\label{KKPushForwardAlongEmbedding}

(\href{ConnesSkandalis84}{Connes-Skandalis 84, above prop. 2.8})

Let $h \colon X \hookrightarrow Y$ be an [[embedding]] of [[compact topological space|compact]] [[smooth manifolds]].

The push-forward constructed from this is supposed to be an element in [[KK-theory]]

\begin{displaymath}
h! \colon KK_d(C(X), C(Y))
\end{displaymath}
in terms of which the push-forward on [[operator K-theory]] is induced by postcomposition:

\begin{displaymath}
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)
 \,,
\end{displaymath}
where $d = dim(X) - dim(Y)$.

Now, if we could ``thicken'' $X$ a bit, namely to a [[tubular neighbourhood]]

\begin{displaymath}
h  \;\colon\; X \hookrightarrow U \stackrel{j}{\hookrightarrow} Y
\end{displaymath}
of $h(X)$ in $Y$ without changing the K-theory of $X$, then the element in question will just be the KK-element

\begin{displaymath}
j! \in KK(C_0(U), C(Y))
\end{displaymath}
induced directly from the [[C\emph{-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}

\begin{displaymath}
\psi \colon C(X) \stackrel{\simeq_{KK}}{\to} C_0(U)
  \,.
\end{displaymath}
The bulk of the technical work in constructing the push-forward is in constructing this equivalence. (\hyperlink{BMRS07}{BMRS 07, example 3.3})

In order for it to exist at all, assume that the [[normal bundle]]

\begin{displaymath}
N_Y X \coloneqq h^\ast(T Y)/ T X
\end{displaymath}
has a [[spin{\tt \symbol{94}}c structure]]. Write $S(N_Y X)$ for the associated [[spinor bundle]].

Then there is an invertible element in [[KK-theory]]

\begin{displaymath}
\iota^X! \in KK_n(C(X), C_0(N_Y X))
\end{displaymath}
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\ldots{}

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

\begin{displaymath}
\Phi \colon U_{h(X)} \hookrightarrow N_Y X
  \,.
\end{displaymath}
This induces a KK-equivalence

\begin{displaymath}
[\Phi] \colon C_0(N_Y X) \stackrel{\simeq_{KK}}{\to} C_0(U)
  \,.
\end{displaymath}
Therefore the push-forward in operator K-theory along $f \colon X \hookrightarrow Y$ is given by postcomposing in [[KK-theory]] with

\begin{displaymath}
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)
  \,.
\end{displaymath}
\hypertarget{KKPushforwardAlongSubmersion}{}\paragraph*{{Along a proper submersion}}\label{KKPushforwardAlongSubmersion}

(\href{ConnesSkandalis84}{Connes-Skandalis 84, above prop. 2.9})

For $\pi \colon X \to Z$ a [[K-orientation|K-oriented]] [[proper map|proper]] [[submersion]] of compact smooth manifolds, the push-forward map along it is reduced to the \hyperlink{KKPushForwardAlongEmbedding}{above} case of an embedding by

\begin{enumerate}%
\item using that by the [[Whitney embedding theorem]] every compact $X$ may be embedded into some $\mathbb{R}^{2q}$ such as to yield an embedding

\begin{displaymath}
h \colon X \to Z \times \mathbb{R}^{2 q}
\end{displaymath}

\item using that there is a KK-equivalence

\begin{displaymath}
\iota^Z! \colon C(Z) \stackrel{\simeq_{KK}}{\to} C_0(Z \times \mathbb{R}^{2q})
  \,.
\end{displaymath}


\end{enumerate}
The resulting push-forward is then given by postcomposition in [[KK-theory]] with

\begin{displaymath}
\pi! \colon C(X) \stackrel{h!}{\to} C_0(Z \times \mathbb{R}^{2}q)
   \underoverset{\simeq_{KK}}{(\iota^Z!)^{-1}}{\to} 
  C(Z)
  \,.
\end{displaymath}
(\hyperlink{BMRS07}{BMRS 07, example 3.4})

\hypertarget{AlongAFibrationOfClosedSpinCManifolds}{}\paragraph*{{Along a smooth fibration of closed $Spin^c$-manifolds}}\label{AlongAFibrationOfClosedSpinCManifolds}

Specifically, for $\pi \colon X \to Z$ a smooth fibration over a closed smooth manifold whose [[fibers]] $X/Z$ are

\begin{itemize}%
\item [[closed manifold|closed]] [[smooth manifolds|smooth]] [[spin{\tt \symbol{94}}c structure|spin{\tt \symbol{94}}c]] [[manifolds]] of even [[dimension]]

\end{itemize}
the push-forward element $\pi! \in KK(C_0(X), C_0(Z))$ is given by the [[Fredholm module|Fredholm]]-[[Hilbert module]] obatined from the fiberwise [[spin{\tt \symbol{94}}c Dirac operator]] acting on the fiberwise [[spinors]]. (\href{ConnesSkandalis84}{Connes-Skandalis 84, proof of lemma 4.7}, \href{BMRS07}{BMRS 07, example 3.9}).

In detail, write

\begin{displaymath}
T(X/Z) \hookrightarrow T X
\end{displaymath}
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{\tt \symbol{94}}c Dirac operator]], acting in each fiber on the [[Hilbert space]] $L^2(X/Z, S_{X/Z})$.

This yields a [[Fredholm module|Fredholm]]-[[Hilbert bimodule]]

\begin{displaymath}
(D_{X/Z}, L^2(X/Z, S_{X/Z}))
\end{displaymath}
which defines an element in [[KK-theory]]

\begin{displaymath}
\pi ! \in KK(C_0(X), C_0(Z))
  \,.
\end{displaymath}
Postcompositon with this is the push-forward map in K/KK-theory, equivalently the [[index]] map of the collection of Dirac operators.

\hypertarget{KKPushforwardAlongGeneralMap}{}\paragraph*{{Along a general K-oriented map}}\label{KKPushforwardAlongGeneralMap}

(\href{ConnesSkandalis84}{Connes-Skandalis 84, def. 2.1})

Now for $f \colon X \to Y$ an arbitray [[K-orientation|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 $Y$:

\begin{displaymath}
f \;\colon\;
  X \stackrel{graph(f)}{\to}
  X \times Y
  \stackrel{p_Y}{\to}
  Y
  \,.
\end{displaymath}
Hence push-forward along such a general map is postcomposition in [[KK-theory]] with

\begin{displaymath}
f! \coloneqq p_Y !\circ graph(f)!
  \,.
\end{displaymath}
(\hyperlink{BMRS07}{BMRS 07, example 3.5})

\hypertarget{KKPushforwardAlongGeneralMap}{}\paragraph*{{In twisted K-theory}}\label{KKPushforwardAlongGeneralMap}

We discuss push forward in K-theory more generally by [[Poincaré duality C\emph{-algebras]] hence [[dual objects]] in [[KK-theory]].}

Let $i \colon Q \to X$ be a map of [[compact topological space|compact]] [[manifolds]] and let $\chi \colon X \to B^2 U(1)$ modulate a [[circle 2-bundle]] regarded as a [[twisted K-theory|twist for K-theory]]. Then forming [[twisted groupoid convolution algebras]] yields a [[KK-theory]] morphism of the form

\begin{displaymath}
C_{i^\ast \chi}(Q)
  \stackrel{i^\ast}{\longleftarrow}
  C_{\chi}(X)
  \,,
\end{displaymath}
with notation as in \href{Poincar&#233;+duality+algebra#CStarAlgebraOf2BundleOnManifold}{this definition}. By \href{Poincar&#233;+duality+algebra#DualOfCompactManifoldWithTwist}{this proposition} the [[dual morphism]] is of the form

\begin{displaymath}
C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\frac{W_3(\tau_X)}{\chi}}(X)
  \,.
\end{displaymath}
If we assume that $X$ has a [[spin{\tt \symbol{94}}c structure]] then this is

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

\begin{displaymath}
i_! \colon K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) \to K_{\bullet -\chi}
  \,.
\end{displaymath}
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 gauge field|background]] [[B-field]] and an element

\begin{displaymath}
[\xi] \in K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q)
\end{displaymath}
is called (the K-class of) a \emph{[[Chan-Paton gauge field]]} on the D-brane satisfying the \emph{[[Freed-Witten-Kapustin anomaly cancellation]]} mechanism. (The orginal \emph{[[Freed-Witten anomaly cancellation]]} assumes $\xi$ given by a [[twisted unitary bundle|twisted line bundle]] in which case it exhibits a [[twisted spin{\tt \symbol{94}}c structure]] on $Q$.) Finally its [[fiber integration|push-forward]]

\begin{displaymath}
[i_! \xi] \in K_{\bullet- \chi}(X)
\end{displaymath}
is called the corresponding \emph{[[D-brane charge]]}.

\hypertarget{of_cohesive_differential_form_data}{}\subsubsection*{{Of cohesive differential form data}}\label{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 \emph{\href{integration+of+differential+forms#InCohesiveHomotopyTypeTheory}{integration of differential forms -- In cohesive homotopy-type theory}}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{to_the_point}{}\subsubsection*{{To the point}}\label{to_the_point}

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

\begin{displaymath}
\int_E:H^\bullet(E)\to H^{\bullet-dim E}(*)
\end{displaymath}
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 $E$. \ldots{}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Grothendieck-Riemann-Roch theorem]]


\item [[transfer context]], [[sheaf with transfer]], [[Becker-Gottlieb transfer]]


\item [[geometric quantization by push-forward]]


\item [[twisted Umkehr map]]

\begin{itemize}%
\item [[fiber integration in K-theory]]

\end{itemize}


\end{itemize}
[[!include generalized fiber integration synonyms - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Frank Adams]], pages 25-27 in \emph{[[Stable homotopy and generalized homology]]}, Chicago Lectures in mathematics, 1974


\item [[Stanley Kochmann]], section 4.3 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996



\end{itemize}
Fiber integration of differential forms is discussed in section VII of volume I of

\begin{itemize}%
\item [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], \emph{[[Connections, Curvature, and Cohomology]]} Academic Press (1973)

\end{itemize}
A quick summary can be found from \href{http://www.math.wisc.edu/~gstgc/slides/Koytcheff.pdf#page=14}{slide 14} on in

\begin{itemize}%
\item [[Robin Koytcheff]], \emph{A homotopy-theoretic view of Bott-Taubes integrals} (\href{http://www.math.wisc.edu/~gstgc/slides/Koytcheff.pdf}{pdf slides})

\end{itemize}
More details are in

\begin{itemize}%
\item [[Ralph Cohen]], [[John Klein]], \emph{Umkehr Maps} (\href{http://arxiv.org/abs/0711.0540}{arXiv:0711.0540})

\end{itemize}
\hypertarget{ReferencesInKKTheory}{}\subsubsection*{{In noncommutative topology and KK-theory}}\label{ReferencesInKKTheory}

Push-forward in [[twisted K-theory]] is discussed in

\begin{itemize}%
\item [[Alan Carey]], [[Bai-Ling Wang]], \emph{Thom isomorphism and Push-forward map in twisted K-theory} (\href{http://arxiv.org/abs/math/0507414}{arXiv:math/0507414})

\end{itemize}
and section 10 of (\hyperlink{ABG10}{ABG, 10})

Discussion of fiber integration [[Gysin maps]]/Umkehr maps in [[noncommutative topology]]/[[KK-theory]] as \hyperlink{InKKTheory}{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

\begin{itemize}%
\item [[Alain Connes]], \emph{A survey of foliations and operator algebras}, Proceedings of the A.M.S., 38, 521-628 (1982) (\href{http://www.alainconnes.org/docs/foliationsfine.pdf}{pdf})

\end{itemize}
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]]:

\begin{itemize}%
\item [[Alain Connes]], [[Georges Skandalis]], \emph{The longitudinal index theorem for foliations}. Publ. Res. Inst. Math. Sci. 20, no. 6, 1139--1183 (1984) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.4218}{web})

\end{itemize}
More on this is in

\begin{itemize}%
\item [[Michel Hilsum]], [[Georges Skandalis]], \emph{Morphismes K-orient\'e{} d'espace de feuille et fonctoralit\'e{} en th\'e{}orie de Kasparov}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 4, 20 no. 3 (1987), p. 325-390 (\href{http://www.numdam.org/item?id=ASENS_1987_4_20_3_325_0}{numdam})

\end{itemize}
(the article that introuced [[Hilsum-Skandalis morphisms]]).

This is further generalized to [[circle 2-bundle]]-twisted convolution algebras of foliations in

\begin{itemize}%
\item Paulo Carrillo Rouse, [[Bai-Ling Wang]], \emph{Twisted longitudinal index theorem for foliations and wrong way functoriality} (\href{http://arxiv.org/abs/1005.3842}{arXiv:1005.3842})

\end{itemize}
Dicussion for general [[C\emph{-algebras]] is in section 3 of}

\begin{itemize}%
\item Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], \emph{Noncommutative correspondences, duality and D-branes in bivariant K-theory}, Adv. Theor. Math. Phys.13:497-552,2009 (\href{http://arxiv.org/abs/0708.2648}{arXiv:0708.2648})

\end{itemize}
and specifically including also [[twisted K-theory]] again (and the relation to [[D-brane charge]]) in section 7 of

\begin{itemize}%
\item Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], \emph{D-Branes, RR-Fields and Duality on Noncommutative Manifolds}, Commun. Math. Phys. 277:643-706,2008 (\href{http://arxiv.org/abs/hep-th/0607020}{arXiv:hep-th/0607020})

\end{itemize}
\hypertarget{abstract_formulation}{}\subsubsection*{{Abstract formulation}}\label{abstract_formulation}

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

\begin{itemize}%
\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], \emph{Twists of K-theory and TMF}, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], \emph{Superstrings, Geometry, Topology, and $C^*$-algebras}, Proceedings of Symposia in Pure Mathematics \href{http://www.ams.org/bookstore-getitem/item=PSPUM-81}{vol 81}, American Mathematical Society (\href{http://arxiv.org/abs/1002.3004}{arXiv:1002.3004})

\end{itemize}
and in section 10 of

\begin{itemize}%
\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], \emph{Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map} (\href{http://arxiv.org/abs/1112.2203}{arXiv:1112.2203})

\end{itemize}
This is reviewed and used also in

\begin{itemize}%
\item [[Joost Nuiten]], \emph{[[schreiber:master thesis Nuiten|Cohomological quantization of boundary prequantum field theory]]}, MSc thesis, Utrecht 2013

\end{itemize}
Formulation of this in [[linear homotopy-type theory]] is discussed in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:Homotopy-type semantics for quantization]]}

\end{itemize}
[[!redirects umkehr map]] [[!redirects umkehr maps]] [[!redirects Umkehr map]] [[!redirects Umkehr maps]]

[[!redirects fiber integration in generalized cohomology]] [[!redirects integration in generalized cohomology]]

[[!redirects push-forward in generalized cohomology]] [[!redirects pushforward in generalized cohomology]]

[[!redirects fiber integral]] [[!redirects fiber integrals]]

[[!redirects fiberwise integral]] [[!redirects fiberwise integrals]]



\end{document}