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\begin{document}

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\section*{stable splitting of mapping spaces}

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

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{goodwillie_calculus}{}\paragraph*{{Goodwillie calculus}}\label{goodwillie_calculus}

[[!include Goodwillie calculus - contents]]

\hypertarget{mapping_space}{}\paragraph*{{Mapping space}}\label{mapping_space}

[[!include mapping space - 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{Statements}{Statements}\dotfill \pageref*{Statements} \linebreak
\noindent\hyperlink{prelude_equivalence_to_the_infinite_configuration_space}{Prelude: Equivalence to the infinite configuration space}\dotfill \pageref*{prelude_equivalence_to_the_infinite_configuration_space} \linebreak
\noindent\hyperlink{StableSplittings}{Stable splitting of mapping spaces}\dotfill \pageref*{StableSplittings} \linebreak
\noindent\hyperlink{InTermsOfGoodwillieTowers}{In terms of Goodwillie-Taylor towers}\dotfill \pageref*{InTermsOfGoodwillieTowers} \linebreak
\noindent\hyperlink{lax_closed_structure_on_}{Lax closed structure on $\Sigma^\infty$}\dotfill \pageref*{lax_closed_structure_on_} \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}

The [[stabilization]]/[[suspension spectrum]] $\Sigma^\infty  Maps(X,A)$ of [[mapping spaces]] $Maps(X,A)$ between suitable [[CW-complexes]] $X, A$ happens to decompose as a [[direct sum]] of [[spectra]] (a [[wedge sum]]) in a useful way, related to the expression of the [[Goodwillie derivatives]] of the functor $Maps(X,-)$ and often expressible in terms of the [[configuration space (mathematics)|configuration spaces]] of $X$.

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

The stable splitting of mapping spaces discussed \hyperlink{StableSplittings}{below} have summands given by [[configuration spaces of points]], or generalizations thereof. To be self-contained, we recall the relevant definitions here.

The following Def. \ref{ConfigurationSpacesOfnPoints} is not the most general definition of [[configuration spaces of points]] that one may consider in this context, instead it is streamlined to certain applications. See Remark \ref{ComparisonToNotationInLiterature} below for comparison of notation used here to notation used elsewhere.

\begin{defn}
\label{ConfigurationSpacesOfnPoints}\hypertarget{ConfigurationSpacesOfnPoints}{}
\textbf{([[configuration spaces of points]])}

Let $X$ be a [[manifold]], possibly with [[manifold with boundary|boundary]].

For $n \in \mathbb{N}$, the \emph{\textbf{configuration space of $n$ distinguishable points} in $X$ disappearing at the boundary} is the [[topological space]]

\begin{equation}
\mathrm{Conf}^{ord}_{n}(X)
    \;\coloneqq\;
      \big(
        X^n \setminus \mathbf{\Delta}_X^n
      \big)
      / \partial(X^n)
\label{DistinguishableConfigurationSpaceJustForX}\end{equation}
which is the [[complement]] of the [[fat diagonal]] $\mathbf{\Delta}_X^n \coloneqq \{(x^i) \in X^n | \underset{i,j}{\exists} (x^i = x^j) \}$ inside the $n$-fold [[product space]] of $X$ with itself, followed by [[quotient space|collapsing]] any configurations with elements on the [[boundary]] of $X$ to a common [[pointed topological space|base point]].

Then the \emph{\textbf{configuration space of $n$ in-distinguishable points} in $X$ is the further [[quotient topological space]]}

\begin{equation}
\mathrm{Conf}_{n}(X)
    \;\coloneqq\;
    Conf_n^{ord}(X)/\Sigma_n
    \;=\;
    \Big(
      \big(
        X^n \setminus \mathbf{\Delta}_X^n
      \big)
      / \partial(X^n)
    \Big)
    /\Sigma(n)
    \,,
\label{ConfigurationSpaceJustForX}\end{equation}
where $\Sigma(n)$ denotes the evident [[action]] of the [[symmetric group]] by [[permutation]] of factors of $X$ inside $X^n$.

More generally, let $Y$ be another [[manifold]], possibly with [[manifold with boundary|boundary]]. For $n \in \mathbb{N}$, the \emph{\textbf{configuration space of $n$ points} in $X \times Y$ vanishing at the boundary and distinct as points in $X$} is the [[topological space]]

\begin{equation}
\mathrm{Conf}_{n}(X,Y)
    \;\coloneqq\;
    \Big(
    \big(
      (
        X^n \setminus \mathbf{\Delta}_X^n
      )
      \times
      Y^n
    \big)
    / \partial(X^n \times Y^n)
    \Big)
    /\Sigma(n)
\label{ConfigurationSpaceWithXAndY}\end{equation}
where now $\Sigma(n)$ denotes the evident [[action]] of the [[symmetric group]] by [[permutation]] of factors of $X \times Y$ inside $X^n \times Y^n \simeq (X \times Y)^n$.

This more general definition reduces to the previous case for $Y = \ast \coloneqq \mathbb{R}^0$ being the point:

\begin{displaymath}
\mathrm{Conf}_n(X)
    \;=\;
    \mathrm{Conf}_n(X,\ast)
    \,.
\end{displaymath}
Finally the \emph{\textbf{configuration space of an arbitrary number of points} in $X \times Y$ vanishing at the boundary and distinct already as points of $X$} is the [[quotient topological space]] of the [[disjoint union space]]

\begin{displaymath}
Conf\left( X, Y\right)
  \;\coloneqq\;
  \left(
  \underset{n \in \mathbb{n}}{\sqcup}
      \big(
      (
        X^n \setminus \mathbf{\Delta}_X^n
      )
      \times
      Y^k
    \big)
    /\Sigma(n)
  \right)/\sim
\end{displaymath}
by the [[equivalence relation]] $\sim$ given by

\begin{displaymath}
\big(
    (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}), (x_n, y_n)
  \big)
  \;\sim\;
  \big(
    (x_1, y_1), \cdots, (x_{n-1}, y_{n-1})
  \big)
  \;\;\;\; \Leftrightarrow
  \;\;\;\; (x_n, y_n) \in \partial (X \times Y)
  \,.
\end{displaymath}
This is naturally a [[filtered topological space]] with filter stages

\begin{displaymath}
Conf_{\leq n}\left( X, Y\right)
  \;\coloneqq\;
  \left(
  \underset{k \in \{1, \cdots, n\}}{\sqcup}
      \big(
      (
        X^k \setminus \mathbf{\Delta}_X^k
      )
      \times
      Y^k
    \big)
    /\Sigma(k)
  \right)/\sim
  \,.
\end{displaymath}
The corresponding [[quotient topological spaces]] of the filter stages reproduces the above configuration spaces of a fixed number of points:

\begin{displaymath}
Conf_n(X,Y)
  \;\simeq\;
  Conf_{\leq n}(X,Y) / Conf_{\leq (n-1)}(X,Y)
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{ComparisonToNotationInLiterature}\hypertarget{ComparisonToNotationInLiterature}{}
\textbf{(comparison to notation in the literature)}

The above Def. \ref{ConfigurationSpacesOfnPoints} is less general but possibly more suggestive than what is considered for instance in \hyperlink{Boedigheimer87}{Bödigheimer 87}. Concretely, we have the following translations of notation:

\begin{displaymath}
\itexarray{
      \text{ here: }
        && 
      \itexarray{ \text{ Segal 73,} \\ \text{ Snaith 74}: }
        &&
      \text{ Bödigheimer 87: }
      \\
      \\
      Conf(\mathbb{R}^d,Y) 
        &=& 
      C_d( Y/\partial Y )
        &=&
      C( \mathbb{R}^d, \emptyset; Y )  
      \\
      \mathrm{Conf}_n\left( \mathbb{R}^d \right)
      & = &
      F_n C_d( S^0 ) / F_{n-1} C_d( S^0 )
      & = &
      D_n\left( \mathbb{R}^d, \emptyset; S^0  \right)
      \\
      \mathrm{Conf}_n\left( \mathbb{R}^d, Y \right)
      & = &
      F_n C_d( Y/\partial Y ) / F_{n-1} C_d( Y/\partial Y )
      & = &
      D_n\left( \mathbb{R}^d, \emptyset; Y/\partial Y  \right)
      \\
      \mathrm{Conf}_n( X ) && &=& D_n\left( X, \partial X; S^0  \right) 
      \\
      \mathrm{Conf}_n( X, Y  ) && &=& D_n\left( X, \partial X; Y/\partial Y  \right)
    }
\end{displaymath}
Notice here that when $Y$ happens to have [[empty space|empty]] [[boundary]], $\partial Y = \emptyset$, then the [[pushout]]

\begin{displaymath}
Y / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast
\end{displaymath}
is $Y$ with a \href{pointed+topological+space#ForgettingAndAdjoiningBasepoints}{disjoint basepoint attached}. Notably for $Y =\ast$ the [[point space]], we have that

\begin{displaymath}
\ast/\partial \ast = S^0
\end{displaymath}
is the [[0-sphere]].

\end{remark}
\hypertarget{Statements}{}\subsection*{{Statements}}\label{Statements}

\hypertarget{prelude_equivalence_to_the_infinite_configuration_space}{}\subsubsection*{{Prelude: Equivalence to the infinite configuration space}}\label{prelude_equivalence_to_the_infinite_configuration_space}

First recall the following equivalence already before [[stabilization]]:

\begin{prop}
\label{ScanningMapEquivalenceOverCartesianSpace}\hypertarget{ScanningMapEquivalenceOverCartesianSpace}{}
For

\begin{enumerate}%
\item $d \in \mathbb{N}$, $d \geq 1$ a [[natural number]] with $\mathbb{R}^d$ denoting the [[Cartesian space]]/[[Euclidean space]] of that [[dimension]],


\item $Y$ a [[manifold]], with [[inhabited set|non-empty]] [[manifold with boundary|boundary]] so that $Y / \partial Y$ is [[connected topological space|connected]],



\end{enumerate}
the [[scanning map]] constitutes a [[homotopy equivalence]]

\begin{displaymath}
Conf\left( 
    \mathbb{R}^d, Y
  \right)
  \overset{scan}{\longrightarrow}
  \Omega^d \Sigma^d (Y/\partial Y)
\end{displaymath}
between

\begin{enumerate}%
\item the configuration space of arbitrary points in $\mathbb{R}^d \times Y$ vanishing at the boundary (Def. \ref{ConfigurationSpacesOfnPoints})


\item the $d$-fold [[loop space]] of the $d$-fold [[reduced suspension]] of the [[quotient space]] $Y / \partial Y$ (regarded as a [[pointed topological space]] with basepoint $[\partial Y]$).



\end{enumerate}
In particular when $Y = \mathbb{D}^k$ is the [[closed ball]] of [[dimension]] $k \geq 1$ this gives a [[homotopy equivalence]]

\begin{displaymath}
Conf\left( 
    \mathbb{R}^d, \mathbb{D}^k
  \right)
  \overset{scan}{\longrightarrow}
  \Omega^d S^{ d + k }
\end{displaymath}
with the $d$-fold [[loop space]] of the [[n-sphere|(d+k)-sphere]].

\end{prop}
(\hyperlink{May72}{May 72, Theorem 2.7}, \hyperlink{Segal73}{Segal 73, Theorem 3})

\hypertarget{StableSplittings}{}\subsubsection*{{Stable splitting of mapping spaces}}\label{StableSplittings}

\begin{prop}
\label{StableSplittingOfMappingSpacesOutOfEuclideanSpace}\hypertarget{StableSplittingOfMappingSpacesOutOfEuclideanSpace}{}
\textbf{([[stable splitting of mapping spaces]] out of [[Euclidean space]]/[[n-spheres]])}

For

\begin{enumerate}%
\item $d \in \mathbb{N}$, $d \geq 1$ a [[natural number]] with $\mathbb{R}^d$ denoting the [[Cartesian space]]/[[Euclidean space]] of that [[dimension]],


\item $Y$ a [[manifold]], with [[inhabited set|non-empty]] [[manifold with boundary|boundary]] so that $Y / \partial Y$ is [[connected topological space|connected]],



\end{enumerate}
there is a [[stable weak homotopy equivalence]]

\begin{displaymath}
\Sigma^\infty Conf(\mathbb{R}^d, Y)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
\end{displaymath}
between

\begin{enumerate}%
\item the [[suspension spectrum]] of the [[configuration space of points|configuration space]] of an arbitrary number of points in $\mathbb{R}^d \times Y$ vanishing at the boundary and distinct already as points of $\mathbb{R}^d$ (Def. \ref{ConfigurationSpacesOfnPoints})


\item the [[direct sum]] (hence: [[wedge sum]]) of [[suspension spectra]] of the [[configuration space of points|configuration spaces]] of a fixed number of points in $\mathbb{R}^d \times Y$, vanishing at the boundary and distinct already as points in $\mathbb{R}^d$ (also Def. \ref{ConfigurationSpacesOfnPoints}).



\end{enumerate}
Combined with the [[stabilization]] of the [[scanning map]] [[homotopy equivalence]] from Prop. \ref{ScanningMapEquivalenceOverCartesianSpace} this yields a [[stable weak homotopy equivalence]]

\begin{equation}
Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y))
  =
  Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y))
  =
  \Omega^d \Sigma^d (Y/\partial Y)
  \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow}
  \Sigma^\infty Conf(\mathbb{R}^d, Y)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
\label{StableSplittingOfMappingSpacesOutOfSphere}\end{equation}
between the latter direct sum and the [[suspension spectrum]] of the [[mapping space]] of pointed [[continuous functions]] from the [[n-sphere|d-sphere]] to the $d$-fold [[reduced suspension]] of $Y / \partial Y$.

\end{prop}
(\hyperlink{Snaith74}{Snaith 74, theorem 1.1}, \hyperlink{Boedigheimer87}{Bödigheimer 87, Example 2})

In fact by \hyperlink{Boedigheimer87}{Bödigheimer 87, Example 5} this equivalence still holds with $Y$ treated on the same footing as $\mathbb{R}^d$, hence with $Conf_n(\mathbb{R}^d, Y)$ on the right replaced by $Conf_n(\mathbb{R}^d \times Y)$ in the well-adjusted notation of Def. \ref{ConfigurationSpacesOfnPoints}:

\begin{displaymath}
Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y))
  =
  Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y))
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)
\end{displaymath}
\hypertarget{InTermsOfGoodwillieTowers}{}\subsubsection*{{In terms of Goodwillie-Taylor towers}}\label{InTermsOfGoodwillieTowers}

We discuss the interpretation of the above stable splitting of mapping spaces from the point of view of [[Goodwillie calculus]], following \hyperlink{Arone99}{Arone 99, p. 1-2}, \hyperlink{Goodwillie03}{Goodwillie 03, p. 6}.

Observe that the [[configuration space of points]] $Conf_n(X,Y)$ from Def. \ref{ConfigurationSpacesOfnPoints}, given by the formula \eqref{ConfigurationSpaceWithXAndY}

\begin{displaymath}
Conf_n(X,Y)
  \;\coloneqq\;
  \Big(
    \big(
      (
        X^n \setminus \mathbf{\Delta}_X^n
      )
      \times
      Y^n
    \big)
    / \partial(X^n \times Y^n)
  \Big)
  /\Sigma(n)
\end{displaymath}
is the [[quotient]] by the [[symmetric group]]-[[action]] of the \emph{[[smash product]]} $Conf_n(X) \wedge (Y/\partial Y)^n$ of the plain Configuration space $Conf_n(X)$ \eqref{ConfigurationSpaceJustForX} (regarded as a [[pointed topological space]] with basepoint the class of the [[boundary]] $\left[\partial\left(X^n\right)\right]$) with the analogous [[pointed topological space]] given by $Y$, the latter in fact being (since here we do not form the [[complement]] by the [[fat diagonal]]) an $n$-fold [[smash product]] itself:

\begin{displaymath}
Y^{\times_n}/\partial (Y^{\times_n}) 
  \;\simeq\;
  ( Y/\partial Y )^{\wedge_n}
  \,.
\end{displaymath}
Hence in summary:

\begin{equation}
Conf_n(X, Y)
  \;\simeq\;
  Conf^{ord}_n(X) \wedge_{\Sigma(n)} \left( Y/\partial Y \right)^{\wedge_n}
  \,,
\label{ConfSplitsAsSmashProduct}\end{equation}
where

\begin{displaymath}
Conf_n^{ord}(X)
  \;\coloneqq\;
  \left(
    X^{\times_n} \setminus \mathbf{\Delta}_X^n
  \right)/ \partial(X^n)
\end{displaymath}
is the ordered configuration space \eqref{DistinguishableConfigurationSpaceJustForX}.

This construction, regarded as a [[functor]] from [[pointed topological spaces]] to [[spectra]]

\begin{displaymath}
\itexarray{
    Top^{\ast/} 
    &\longrightarrow&
    Spectra
    \\
    Z 
      &\mapsto&
    \Sigma^\infty Conf^{ord}_n(X) \wedge_{\Sigma(n)} Z^{\wedge_n}
  }
\end{displaymath}
is an [[n-homogeneous (∞,1)-functor]] in the sense of [[Goodwillie calculus]], and hence the partial [[wedge sums]] as $n$ ranges

\begin{equation}
Z 
    \;\mapsto\;
  \underset{k \in \{1, \cdot, n\}}{\bigoplus}
  \Sigma^\infty Conf^{ord}_k(X) \wedge_{\Sigma(k)} Z^{\wedge_k}
\label{IdentifyingTheGoodwillieTaylorStage}\end{equation}
are [[n-excisive (∞,1)-functors]]. Moreover, by the stable splitting of mapping spaces \eqref{StableSplittingOfMappingSpacesOutOfSphere} of Prop. \ref{StableSplittingOfMappingSpacesOutOfEuclideanSpace}, there is a [[projection]] morphism onto the first $n$ [[wedge sum|wedge summands]]

\begin{equation}
\itexarray{
    Maps_{cp}(\mathbb{R}^d, \Sigma^d Z)
    &=&
    Maps^{\ast/}( S^d, \Sigma^d Z)
    &\simeq&
    \underset{k \in \mathbb{N}}{\oplus} 
      \Sigma^\infty Conf^{ord}_k(\mathbb{R}^d) \wedge_{\Sigma(k)} Z^{\wedge_k}
    \\
    &&
    &&
    \Big\downarrow {}^{\mathrlap{ p_n }}
    \\
    &&
    &&
     \underset{k \in \{1, \cdot, n\}}{\bigoplus}
     \Sigma^\infty Conf^{ord}_k( \mathbb{R}^d ) \wedge_{\Sigma(k)} Z^{\wedge_k}
  }
\label{ProjectionMaps}\end{equation}
and this is [[n-connected morphism|(n+1)k-connected]] when $Z$ is [[n-connected object|k-connected]].

By [[Goodwillie calculus]] this means that \eqref{IdentifyingTheGoodwillieTaylorStage} are, up to [[equivalence in an (infinity,1)-category|equivalence]], the stages

\begin{equation}
P_n Maps^{\ast/}( S^d, \Sigma^d (-))
  \;\colon\;
  Z \mapsto
  \underset{k \in \{1, \cdot, n\}}{\bigoplus}
  \Sigma^\infty Conf^{ord}_k(S^d, Z)
\label{TheGoodwillieStagesOfTheMappingSpaceFunctor}\end{equation}
at $Z \in Top^{\ast/}$ of the [[Goodwillie-Taylor tower]] for the [[mapping space]]-functor

\begin{displaymath}
Maps_{cp}(\mathbb{R}^d, \Sigma^d (-))
  =
  Maps^{\ast/}( S^d, \Sigma^d (-))
  \;\colon\;
  Top^{\ast/} \longrightarrow Top^{\ast/}
  \,.
\end{displaymath}
Therefore the stable splitting theorem \ref{StableSplittingOfMappingSpacesOutOfEuclideanSpace} may equivalently be read as expressing the mapping space functor equivalently as the [[limit]] over its [[Goodwillie-Taylor tower]].

(\hyperlink{Arone99}{Arone 99, p. 1-2}, \hyperlink{Goodwillie03}{Goodwillie 03, p. 6})

$\,$

\hypertarget{lax_closed_structure_on_}{}\subsubsection*{{Lax closed structure on $\Sigma^\infty$}}\label{lax_closed_structure_on_}

Notice that the first stage in the [[Goodwillie-Taylor tower]] of $Maps(S^d, \Sigma^d(-))$ is

\begin{displaymath}
\begin{aligned}
    P_1 Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y) )
    & =
    \Sigma^\infty Conf^{ord}_1( \mathbb{R}^d , Y )
    \\
    & \simeq
    \Sigma^\infty 
      \underset{\simeq S^0}{\underbrace{Conf^{ord}_1( \mathbb{R}^d )}} 
      \wedge (Y/\partial Y)
    \\
    & \simeq
     \Sigma^\infty (Y/\partial Y)
    \\
    & \simeq 
     \Omega^d \Sigma^d  \Sigma^\infty (Y/\partial Y)
    \\
    & \simeq
    Maps\left(  \Sigma^\infty S^d, \Sigma^d (Y/\partial Y) \right)
  \end{aligned}
\end{displaymath}
Here in the first step we used \eqref{TheGoodwillieStagesOfTheMappingSpaceFunctor}, in the second step we used \eqref{ConfSplitsAsSmashProduct}. Under the brace we observe that space of configurations of a single point in $\mathbb{R}^d$ is trivially $\mathbb{R}^d$ itself, which is [[contractible topological space|contractible]] $\mathbb{R}^d \simeq \ast$ and, due to [[empty set|empty]] [[boundary]] of $\mathbb{R}^d$, contributes a [[0-sphere]]-factor to the [[smash product]], which disappears. In the last last two steps we trivially rewrote the result to exhibit it as a [[mapping spectrum]].

Therefore the projection $p_1$ \eqref{ProjectionMaps} to the first stage of the [[Goodwillie-Taylor tower]] is of the form

\begin{displaymath}
p_1
  \;\colon\;
  \Sigma^\infty Maps\left(  S^d , \Sigma^d (Y /\partial Y) \right)
  \longrightarrow
  Maps
  \left(  
    \Sigma^\infty S^d, \Sigma^\infty \Sigma^d (Y / \partial Y)
  \right)
  \,.
\end{displaymath}
Since $\Sigma^\infty$ is a [[strong monoidal functor]] (\href{suspension+spectrum#StrongMonoidalness}{here}), there is a canonical comparison morphism of this form, exhibiting the induce [[closed functor|lax closed]]-structure on $\Sigma^\infty$. Probably $p_1$ coincides with that canonical morphism, up to equivalence.

\begin{quote}%
Does it?


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

\begin{itemize}%
\item [[mapping spectrum]]

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

The theorem is originally due to

\begin{itemize}%
\item [[Victor Snaith]], \emph{A stable decomposition of $\Omega^n S^n X$}, Journal of the London Mathematical Society 7 (1974), 577 - 583 (\href{https://www.maths.ed.ac.uk/~v1ranick/papers/snaiths.pdf}{pdf})

\end{itemize}
using the homotopy equivalence before stabilization due to

\begin{itemize}%
\item [[Peter May]], \emph{The geometry of iterated loop spaces}, Springer 1972 (\href{https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf}{pdf})


\item [[Graeme Segal]], \emph{Configuration-spaces and iterated loop-spaces}, Invent. Math. \textbf{21} (1973), 213--221. MR 0331377 (\href{http://dodo.pdmi.ras.ru/~topology/books/segal.pdf}{pdf})



\end{itemize}
An alternative proof is due to

\begin{itemize}%
\item [[Ralph Cohen]], \emph{Stable proof of stable splittings}, Math. Proc. Camb. Phil. Soc., 1980, 88, 149 (\href{https://doi.org/10.1017/S030500410005742X}{doi:10.1017/S030500410005742X}, \href{https://www.cambridge.org/core/services/aop-cambridge-core/content/view/247D9F951F8AB99000E4FF6CB2DB9EA2/S030500410005742Xa.pdf/div-class-title-stable-proofs-of-stable-splittings-div.pdf}{pdf})

\end{itemize}
Review and generalization is due to

\begin{itemize}%
\item [[Carl-Friedrich Bödigheimer]], \emph{Stable splittings of mapping spaces}, Algebraic topology. Springer 1987. 174-187 (\href{http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf}{pdf})

\end{itemize}
Interpretation in terms of the [[Goodwillie-Taylor tower]] of mapping spaces is due to

\begin{itemize}%
\item [[Greg Arone]], \emph{A generalization of Snaith-type filtration}, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. (\href{https://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02405-8/S0002-9947-99-02405-8.pdf}{pdf})


\item [[Michael Ching]], \emph{Calculus of Functors and Configuration Spaces}, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 (\href{https://www3.amherst.edu/~mching/Work/skye.pdf}{pdf})


\item [[Thomas Goodwillie]], p. 6 of \emph{Calculus. III. Taylor series}, Geom. Topol. 7 (2003), 645--711 (\href{http://www.msp.warwick.ac.uk/gt/2003/07/p019.xhtml}{journal}, \href{http://arxiv.org/abs/math/0310481}{arXiv:math/0310481}))



\end{itemize}
Generalization via [[nonabelian Poincaré duality]] from [[codomains]] which are $n$-fold [[suspensions]] to general [[n-connective spaces]]:

\begin{itemize}%
\item Lauren Bandklayder, \emph{Stable splitting of mapping spaces via nonabelian Poincaré duality} (\href{https://arxiv.org/abs/1705.03090}{arxiv:1705.03090})

\end{itemize}
[[!redirects stable splittings of mapping spaces]]



\end{document}