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\begin{document}

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\section*{Pontrjagin-Thom collapse map}

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

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - 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{ComponentDefinitionInTopologicalSpaces}{Component definition in topological spaces}\dotfill \pageref*{ComponentDefinitionInTopologicalSpaces} \linebreak
\noindent\hyperlink{AbstractDefinitionInTermsOfDuality}{Abstract definition in terms of duality}\dotfill \pageref*{AbstractDefinitionInTermsOfDuality} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_between_cohomotopy_and_cobordism}{Relation between cohomotopy and cobordism}\dotfill \pageref*{relation_between_cohomotopy_and_cobordism} \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}

Given an [[embedding of topological spaces|embedding]] of [[smooth manifolds]] $i \colon X \hookrightarrow Y$ of [[codimension]] $n$, the \emph{Thom collapse map} (\hyperlink{Thom54}{Thom 54}) is the [[continuous function]] from $X$ to the [[n-sphere]] which assigns \textbf{asymptotic normal distance} from the [[submanifold]], measured

\begin{enumerate}%
\item in [[direction vector|direction]] [[orthogonality|perpendicular]] to the submanifold, with respect to a [[normal framing]];


\item asymptotically, regarding all points outside a [[tubular neighbourhood]] as being [[one-point compactification|at infinity]].



\end{enumerate}
\begin{quote}%
graphics grabbed from \href{Cohomotopy+charge#SatiSchreiber19}{SS 19}


\end{quote}
For maximal codimension $n$, hence for 0-dimensional [[submanifolds]], hence for [[configuration space of points|configurations of points]], this is alternatively known as the ``electric field map'' (\href{cohomotopy+charge#Salvatore01}{Salvatore 01} following \href{cohomotopy+charge#Segal73}{Segal 73, Section 1}, see also \href{cohomotopy+charge#Knudsen18}{Knudsen 18, p. 49}) or the ``scanning map'' (\href{cohomotopy+charge#Kallel98}{Kallel 98}).

The [[homotopy class]] of the Thom collpase map may be regarded as the \emph{[[Cohomotopy charge]]} of the submanifolds, as measured in $n$-[[Cohomotopy]]-[[generalized cohomology|cohomology theory]].

The PT collapse is a useful approximation to the would-be [[left inverse]] of the [[embedding of topological spaces]]

As such, it is is used to define pushforward of [[cohomology]]-classes along $i$ (``[[Umkehr maps]]''). It also appears as the key step in [[Thom's theorem]].

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

\hypertarget{ComponentDefinitionInTopologicalSpaces}{}\subsubsection*{{Component definition in topological spaces}}\label{ComponentDefinitionInTopologicalSpaces}

All [[topological spaces]] in the following are taken to be [[compact space|compact]].

Consider $X$ and $Y$ two [[manifolds]] and

\begin{displaymath}
i \colon X \hookrightarrow Y
\end{displaymath}
an [[embedding]].

Write

\begin{itemize}%
\item $N_i X \coloneqq i^* T Y/ T X$ for the [[normal bundle]];


\item $Th(N_i X)$ for the [[Thom space]] of the normal bundle;


\item $f \colon N_i X \longrightarrow Y$ for any choice of [[tubular neighbourhood]] of $i$.



\end{itemize}
\begin{defn}
\label{CollapseMap}\hypertarget{CollapseMap}{}
The \textbf{collapse map} (or the \emph{Pontrjagin-Thom construction}) associated to $i$ and the choice of tubular neighbourhood $f$ is

\begin{displaymath}
c_i 
    \colon 
  Y 
    \to 
  Y/(Y - f(N_i X)) 
    \stackrel{\simeq}{\to}
  Th(N_i X)
  \,,
\end{displaymath}
where the first morphism is the [[projection]] onto the [[quotient]] and the second is the canonical [[homeomorphism]] to the [[Thom space]] of the [[normal bundle]].

\end{defn}
\begin{defn}
\label{RefinedCollapseMap}\hypertarget{RefinedCollapseMap}{}
Since in the construction of remark \ref{CollapseMap} every point of $N_i X$ is associated to a particular point of $X$, the collapse map lifts to a map

\begin{displaymath}
Y 
    \longrightarrow 
  X_+ \wedge Th(N_i X)
\end{displaymath}
from $Y$ to the [[smash product]] of the [[Thom space]] (canonically regarded as a [[pointed topological space]]) and the topological space $X$ with a base point adjoined.

\end{defn}
(e.g. \hyperlink{Rudyak98}{Rudyak 98, p. 317})

\begin{example}
\label{ForEmbeddingsIntoAnNSphere}\hypertarget{ForEmbeddingsIntoAnNSphere}{}
Of particular interest is the case where $Y$ in the above is a [[Cartesian space]] $\mathbb{R}^{dim X + k}$ or rather its [[one-point compactification]], the [[n-sphere|sphere]] $S^{dim X + k}$. By the [[Whitney embedding theorem]], for every $n \in \mathbb{N}$ there exists an $k \in \mathbb{N}$ such that every [[manifold]] $X$ of [[dimension]] $n$ has an [[embedding]] $X \hookrightarrow \mathbb{R}^{n+k} \to S^{n+k}$. In this case the collapse map of def. \ref{CollapseMap} has the form

\begin{displaymath}
S^{n+k} \longrightarrow Th(N_i X)
  \,.
\end{displaymath}
Composing this further with the canonical map $N_i X \longrightarrow E O(k) \underset{O(k)}{\times} \mathbb{R}^{k}$ to the universal vector bundle of rank $k$ yields a map

\begin{displaymath}
S^{n+k} \longrightarrow M O(k)
\end{displaymath}
from to the $k$th space in the [[Thom spectrum]] $M O$. This hence defines an element in the [[homotopy group of a spectrum|homotopy group]] $\pi_{k}(M O)$ of the [[Thom spectrum]]. [[Thom's theorem]] says that all elements in the homotopy groups of $M O$ arise this way, and that they retain precisely the information of the [[cobordism]] [[equivalence class]] of manifolds $X$.

In this case the refined Thom collapse map of def. \ref{RefinedCollapseMap} is of the form

\begin{displaymath}
S^{n+k}
    \longrightarrow 
  X_+ \wedge Th(N_i X)
  \,.
\end{displaymath}
\end{example}
\begin{remark}
\label{}\hypertarget{}{}
The refined map in example \ref{ForEmbeddingsIntoAnNSphere} lifts to a morphism of [[spectra]]

\begin{displaymath}
\mathbb{S} \longrightarrow \Sigma_+^\infty X \wedge \Sigma^{-n-k} Th(N_i X)
\end{displaymath}
where $\mathbb{S}$ denotes the [[sphere spectrum]] and $Th(N_i X)$ now the [[Thom spectrum]] of the normal bundle.

This morphism is the [[unit of an adjunction]] which exhibits the [[suspension spectrum]] $\Sigma_+^\infty X$ as a [[dualizable object]] in the [[stable homotopy category]], with [[dual object]] $\Sigma^{-n-k} Th(N_i X)$. See at \emph{[[Atiyah duality]]} and at \emph{[[n-duality]]}.

\end{remark}
Equivalently, one may proceed as follows. For a framed manifold i.e. a manifold $M^n$ with a chosen trivialization of the normal bundle $N_i (M^n)$ in some $\mathbf{R}^{n+r}$ one has $T N_i(M^n)\cong \Sigma^r(M^n_+)$ where $M^n_+$ is the union of $M^n$ with a disjoint base point. Identify a sphere $S^{n+r}$ with a one-point compactification $\mathbf{R}^{n+r}\cup \{\infty\}$. Then the Pontrjagin-Thom construction is the map $S^{n+r}\to Th(N_i X)$ obtained by collapsing the complement of the interior of the unit disc bundle $D(N_i M^n)$ to the point corresponding to $S(N_i M^n)$ and by mapping each point of $D(N_i M^n)$ to itself. Thus to a framed manifold $M^n$ one associates the composition

\begin{displaymath}
S^{n+r}\to Th(N_i X)\cong \Sigma^r M^n_+\to S^r
\end{displaymath}
and its homotopy class defines an element in $\pi_{n+r}(S^r)$.

\hypertarget{AbstractDefinitionInTermsOfDuality}{}\subsubsection*{{Abstract definition in terms of duality}}\label{AbstractDefinitionInTermsOfDuality}

The following is a more abstract description of Pontryagin-Thom collapse in the [[stable homotopy theory]] of [[sphere spectrum]]-[[(∞,1)-module bundles]].

\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 [[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{properties}{}\subsection*{{Properties}}\label{properties}

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

\begin{prop}
\label{}\hypertarget{}{}
For given $i$ all collapse maps for different choices of [[tubular neighbourhood]] $f$ are [[homotopy|homotopic]].

\end{prop}
\begin{proof}
By the fact that the space of [[tubular neighbourhood]]s (see there for details) is [[contractible]].

\end{proof}
\hypertarget{relation_between_cohomotopy_and_cobordism}{}\subsubsection*{{Relation between cohomotopy and cobordism}}\label{relation_between_cohomotopy_and_cobordism}

For $X$ a [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] $D$, the [[Pontryagin-Thom construction]] (e.g. \hyperlink{Kosinski93}{Kosinski 93, IX.5}) identifies the [[set]]

\begin{displaymath}
SubMfd_{/bord}^{d}(X)
\end{displaymath}
of [[cobordism classes]] of [[closed manifold|closed]] and [[normally framed submanifolds]] $\Sigma \overset{\iota}{\hookrightarrow} X$ of [[dimension]] $d$ inside $X$ with the [[cohomotopy]] $\pi^{D-d}(X)$ of $X$ in degree $D- d$

\begin{displaymath}
SubMfd_{/bord}^{d}(X)
    \underoverset{\simeq}{PT}{\longrightarrow}
  \pi^{D-d}(X)
  \,.
\end{displaymath}
(e.g. \hyperlink{Kosinski93}{Kosinski 93, IX Theorem (5.5)})

In particular, by this [[bijection]] the canonical [[group]] [[structure]] on [[cobordism groups]] in sufficiently high [[codimension]] (essentially given by [[disjoint union]] of [[submanifolds]]) this way induces a group structure on the cohomotopy sets in sufficiently high degree.

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

\begin{itemize}%
\item [[Thom's theorem]]


\item [[Thom isomorphism]]



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

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

\begin{itemize}%
\item [[René Thom]], \emph{Quelques propri\'e{}t\'e{}s globales des vari\'e{}t\'e{}s diff\'e{}rentiables} Comment. Math. Helv. 28, (1954). 17-86 (\href{http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259}{digiz:GDZPPN002056259})


\item [[Lev Pontrjagin]], \emph{Smooth manifolds and their applications in Homotopy theory}, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (\href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/pont4.pdf}{pdf}, \href{https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001}{doi:10.1142/9789812772107\_0001})


\item [[John Milnor]], section 7 of \emph{Topology -- From the differentiable viewpoint}, 1965 (\href{http://teachingdm.unito.it/paginepersonali/sergio.console/Dispense/Milnor%20Topology%20from%20%23681EA.pdf}{pdf})


\item [[Stanley Kochmann]], section 1.5 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Yuli Rudyak]], \emph{In Thom spectra, Orientability and Cobordism}, Springer 1998 (\href{http://www.maths.ed.ac.uk/~aar/papers/rudyakthom.pdf}{pdf})


\item [[Cary Malkiewich]], \emph{Unoriented cobordism and MO}, 2011 (\href{http://math.uiuc.edu/~cmalkiew/cobordism.pdf}{pdf})



\end{itemize}
An illustration is given on \href{http://www.math.wisc.edu/~gstgc/slides/Koytcheff.pdf#page=15}{slide 15}

\begin{itemize}%
\item [[Antoni Kosinski]], chapter IX of \emph{Differential manifolds}, Academic Press 1993 (\href{http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf}{pdf})


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


\item [[Victor Snaith]], \emph{Stable homotopy around the arf-Kervaire invariant}, Birkhauser 2009



\end{itemize}
The general abstract formulation in [[stable homotopy theory]] is in 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 is 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}
with an emphases on [[parameterized spectra]].

[[!redirects Pontrjagin-Thom construction]] [[!redirects Pontryagin-Thom construction]]

[[!redirects Pontryagin-Thom collaps map]] [[!redirects Pontrjagin-Thom collaps map]] [[!redirects Pontrjagin-Thom collapse map]]

[[!redirects Thom collapse map]] [[!redirects Thom collapse maps]] [[!redirects Thom collaps map]] [[!redirects Thom collaps maps]]

[[!redirects Pontrjagin-Thom collapse]] [[!redirects Pontryagin-Thom collapse]] [[!redirects Thom collapse]]

[[!redirects Pontryagin-Thom collapse map]]

[[!redirects Pontryagin-Thom collapse maps]] [[!redirects Pontrjagin-Thom collapse maps]]

[[!redirects Pontrjagin-Thom collapse construction]] [[!redirects Pontrjagin-Thom collapse constructions]] [[!redirects Pontryagin-Thom collapse construction]] [[!redirects Pontryagin-Thom collapse constructions]]



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