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\begin{document}

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\section*{twisted Umkehr map}

\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{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{umkehr_map}{Umkehr map}\dotfill \pageref*{umkehr_map} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For $E$ a [[cohomology theory]], and $f \colon X \to Y$ a map of suitable [[spaces]], an ordinary [[Umkehr map]] for the induced map $E^\bullet(f) \colon E^\bullet(Y) \to E^{\bullet}(X)$ is a [[dual morphism]] together with self-[[dual objects|duality]] [[equivalences]] for $E^\bullet(X)$ and $E^\bullet(Y)$ ([[orientation in generalized cohomology|orientation]]/[[Atiyah duality]]+[[Thom isomorphism]]).

More generally, $E^\bullet(X)$ may not be self-dual, but its [[dual object]] may be [[twisted cohomology]] $E^{\bullet+ \chi}(X)$ for some twist $\chi$. In this case the [[Umkehr map]] goes not between the original spaces and their cohomology, but between [[twisted cohomology]] variants of these.

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

\hypertarget{abstract_duality_and_atiyahmilnorspanier_duality__pontryaginthom_collapse}{}\subsubsection*{{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}).

\hypertarget{umkehr_map}{}\subsubsection*{{Umkehr map}}\label{umkehr_map}

\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 [[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 \textbf{twisted Umkehr map}.

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

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

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


\item [[fiber integration in ordinary differential cohomology]]


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

For a detailed discussion of an example in [[K-theory]] see also at \emph{[[Poincaré duality algebra]]} and at \emph{[[Freed-Witten-Kapustin anomaly]]}.


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



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

[[!include generalized fiber integration synonyms - table]]

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

Twisted Umkehr maps in [[topological K-theory]] are discussed (somewhat implicitly sometimes) in the literature on [[KK-theory]]. See the references at \emph{[[Poincaré duality algebra]]}.

The general abstract formulation in [[stable homotopy theory]] 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}
A review and applications to [[quantization]] of [[local prequantum field theory]] is in

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

\end{itemize}
Formalization in [[dependent linear type theory]] is discussed

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:Quantization via Linear homotopy types]]}

\end{itemize}
[[!redirects twisted Umkehr maps]]



\end{document}