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\section*{J-homomorphism}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

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

[[!include stable homotopy theory - 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{OnGroups}{On groups}\dotfill \pageref*{OnGroups} \linebreak
\noindent\hyperlink{delooped_on_classifying_spaces_and_ktheory_classes}{Delooped: On classifying spaces and K-theory classes}\dotfill \pageref*{delooped_on_classifying_spaces_and_ktheory_classes} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{Image}{Image of the J-homomorphism}\dotfill \pageref*{Image} \linebreak
\noindent\hyperlink{ImageOfJTradtionalFormulation}{Traditional formulation}\dotfill \pageref*{ImageOfJTradtionalFormulation} \linebreak
\noindent\hyperlink{description_of_the_image}{Description of the image}\dotfill \pageref*{description_of_the_image} \linebreak
\noindent\hyperlink{characterization_via_the_adams_operations}{Characterization via the Adams operations}\dotfill \pageref*{characterization_via_the_adams_operations} \linebreak
\noindent\hyperlink{the_jspectrum}{The J-spectrum}\dotfill \pageref*{the_jspectrum} \linebreak
\noindent\hyperlink{ImageOfJInChromotopy}{Formulation in chromatic homotopy theory}\dotfill \pageref*{ImageOfJInChromotopy} \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{the_image_of_j}{The image of J}\dotfill \pageref*{the_image_of_j} \linebreak
\noindent\hyperlink{equivariant_enhancement}{Equivariant enhancement}\dotfill \pageref*{equivariant_enhancement} \linebreak
\noindent\hyperlink{relation_to_action_on_general_spectra}{Relation to $O$-action on general spectra}\dotfill \pageref*{relation_to_action_on_general_spectra} \linebreak


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

The \emph{$J$-homomorphism} is traditionally a family of [[group]] [[homomorphisms]]

\begin{displaymath}
J_i \;\colon\; \pi_i(O(n)) \longrightarrow \pi_{n+i}(S^n)
\end{displaymath}
from the [[homotopy groups]] of (the [[topological space]] underlying) the [[orthogonal group]] to the [[homotopy groups of spheres]]. This refines to a morphism of [[∞-groups]]

\begin{displaymath}
J \;\colon\; O \longrightarrow GL_1(\mathbb{S})
\end{displaymath}
from the [[stable orthogonal group]] (regarded as a [[group object in an (∞,1)-category|group object]] in $L_{whe} Top \simeq$ [[∞Grpd]]) to the [[∞-group of units]] of the [[sphere spectrum]], regarded as an [[E-∞ ring spectrum]].

By postcomposition, the [[delooping]] of the J-homomorphism

\begin{displaymath}
B J \;\colon\; B O \to B GL_1(\mathbb{S})
\end{displaymath}
sends real [[vector bundles]] to [[sphere bundles]], namely to [[(∞,1)-line bundles]] with typical [[fiber]] the [[sphere spectrum]] $\mathbb{S}$. See also at \emph{[[Thom space]]} for more on this.

The description of the image of the $J$-homomorphism in the [[stable homotopy groups of spheres]] was an important precursor to the development of [[chromatic homotopy theory]], which is used to explain the periodicities seen in the image of the J-homomorphism (see also \hyperlink{Lurie}{Lurie 10, remark 8}). See also at \emph{[[periodicity theorem]]}.

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

\hypertarget{OnGroups}{}\subsubsection*{{On groups}}\label{OnGroups}

\begin{defn}
\label{SphereAsCompactification}\hypertarget{SphereAsCompactification}{}
For $n \in \mathbb{N}$ regard the $n$-[[sphere]] (as a [[topological space]]) as the [[one-point compactification]] of the [[Cartesian space]] $\mathbb{R}^n$

\begin{displaymath}
S^n \simeq (\mathbb{R}^n)^\ast
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{ActionOfOrthogonalOnSphere}\hypertarget{ActionOfOrthogonalOnSphere}{}
Since the process of [[one-point compactification]] is a [[functor]] on [[proper maps]], hence on [[homeomorphisms]], via def. \ref{SphereAsCompactification} the $n$-sphere inherits from the canonical [[action]] of the [[orthogonal group]] $O(n)$ on $\mathbb{R}^n$ an [[action]]

\begin{displaymath}
O(n) \times S^n \longrightarrow S^n
\end{displaymath}
(by [[continuous maps]]) which preserves the base point (the ``point at infinity'').

\end{remark}
For definiteness we distinguish in the following notationally between

\begin{enumerate}%
\item the $n$-[[sphere]] $S^n \in Top$ regarded as a [[topological space]];


\item its [[homotopy type]] $\Pi(S^n) \in L_{whe} Top \simeq$ [[∞Grpd]] given by its [[fundamental ∞-groupoid]].



\end{enumerate}
Similarly we write $\Pi(O(n))$ for the [[homotopy type]] of the [[orthogonal group]], regarded as a [[group object in an (∞,1)-category]] in [[∞Grpd]] (using that the [[shape modality]] $\Pi$ preserves [[finite products]]).

\begin{defn}
\label{AutoequivalencesOfnSphere}\hypertarget{AutoequivalencesOfnSphere}{}
For $n \in \mathbb{N}$ write $H(n)$ for the [[automorphism ∞-group]] of homotopy self-equivalences $S^n \longrightarrow S^n$, hence

\begin{displaymath}
H(n) \coloneqq Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The [[∞-group]] $H(n)$, def. \ref{AutoequivalencesOfnSphere}, constitutes the two [[connected components]] of the $n$-fold based [[loop space]] $\Omega^n S^n$ corresponding to the [[homotopy groups]] $\pm 1 \in \pi_n(S^n)$.

\end{remark}
\begin{defn}
\label{OrthogonalActionOnSphereOnHomotopyGroups}\hypertarget{OrthogonalActionOnSphereOnHomotopyGroups}{}
Via the presentation of [[∞Grpd]] by the [[cartesian closed model structure|cartesian closed]] [[model structure on topological spaces|model structure on compactly generated topological spaces]] (and using that $S^n$ and $O(n)$ and hence their product are [[compact topological spaces|compact]]) we have that for $n \in \mathbb{N}$ the [[continuous function|continuous]] [[action]] of $O(n)$ on $S^n$ of remark \ref{ActionOfOrthogonalOnSphere}, which by [[cartesian closed category|cartesian closure]] is equivalently a homomorphism of [[topological groups]] of the form

\begin{displaymath}
O(n) \longrightarrow Aut_{Top^{\ast/}}(S^n)
  \,,
\end{displaymath}
induces a homomorphism of [[∞-groups]] of the form

\begin{displaymath}
\Pi(O(n)) \longrightarrow Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
  \,.
\end{displaymath}
This in turn induces for each $i \in \mathbb{N}$ homomorphisms of [[homotopy groups]] of the form

\begin{displaymath}
\pi_i(O(n)) \longrightarrow \pi_i(\Omega^n S^n) \simeq \pi_{n+i}(S^n)
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{OrthogonalActionOnSphereOnHomotopyGroups}\hypertarget{OrthogonalActionOnSphereOnHomotopyGroups}{}
By construction, the homomorphisms of remark \ref{OrthogonalActionOnSphereOnHomotopyGroups} are compatible with [[suspension]] in that for all $n \in \mathbb{N}$ the [[diagrams]]

\begin{displaymath}
\itexarray{
    O(n) &\longrightarrow& Aut_{Top^{\ast/}}(S^n)
    \\
    \downarrow && \downarrow
    \\
    O(n+1) &\longrightarrow& Aut_{Top^{\ast/}}(S^{n+1})
  }
\end{displaymath}
in $Grp(Top)$ commute, and hence so do the diagrams

\begin{displaymath}
\itexarray{
    \Pi(O(n)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
    \\
    \downarrow && \downarrow
    \\
    \Pi(O(n+1)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^{n+1}))
  }
\end{displaymath}
in $Grp(\infty Grpd)$, up to [[homotopy]].

\end{remark}
Therefore one can take the [[direct limit]] over $n$:

\begin{defn}
\label{JHom}\hypertarget{JHom}{}
By remark \ref{OrthogonalActionOnSphereOnHomotopyGroups} there is induced a homomorphism

\begin{displaymath}
J_i \;\colon\; \pi_\bullet(O) \longrightarrow \pi_\bullet(\mathbb{S})
\end{displaymath}
from the [[homotopy groups]] of the [[stable orthogonal group]] to the [[stable homotopy groups of spheres]]. This is called the \textbf{J-homomorphism}.

\end{defn}
\hypertarget{delooped_on_classifying_spaces_and_ktheory_classes}{}\subsubsection*{{Delooped: On classifying spaces and K-theory classes}}\label{delooped_on_classifying_spaces_and_ktheory_classes}

\begin{remark}
\label{DeloopedJ}\hypertarget{DeloopedJ}{}
Since the maps of def. \ref{OrthogonalActionOnSphereOnHomotopyGroups} are [[∞-group]] [[homomorphisms]], there exists their [[delooping]]

\begin{displaymath}
B J \;\colon\; B O \longrightarrow B GL_1(\mathbb{S}) = B H
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Here $GL_1(\mathbb{S})$ is the [[∞-group of units]] of the [[sphere spectrum]].

\end{remark}
This map $B J$ is the [[universal characteristic class]] of stable [[vector bundles]] with values in [[spherical fibrations]]:

\begin{defn}
\label{SphereBundleOfVectorBundle}\hypertarget{SphereBundleOfVectorBundle}{}
For $V \to X$ a [[vector bundle]], write $S^V$ for its [[fiber]]-wise [[one-point compactification]]. This is a [[sphere bundle]]/[[spherical fibration]]. Write $\mathbb{S}^V$ for the $X$-[[parameterized spectrum]] which is fiberwise the [[suspension spectrum]] of $S^V$.

\end{defn}
It is immediate that:

\begin{prop}
\label{SphericalFibrationsOfVectorBundlesClassifiedViaJ}\hypertarget{SphericalFibrationsOfVectorBundlesClassifiedViaJ}{}
\textbf{([[spherical fibrations]] of [[vector bundles]] classified via J-homomorphism)}

For $V \to X$ a [[vector bundle]] classified by a map $X \to B O$, the corresponding [[spherical fibration]] $\mathbb{S}^V$, def. \ref{SphereBundleOfVectorBundle}, is classified by $X \to B O \stackrel{B J}{\longrightarrow} B GL_1(\mathbb{S})$, def. \ref{DeloopedJ}.

\end{prop}
This construction descends to a map

\begin{displaymath}
KO^0(X) \longrightarrow Sph(X)
\end{displaymath}
from [[topological K-theory]] to [[spherical fibrations]]

(\ldots{}) (\href{http://mathoverflow.net/a/156369/381}{MO discussion})

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{Image}{}\subsubsection*{{Image of the J-homomorphism}}\label{Image}

\hypertarget{ImageOfJTradtionalFormulation}{}\paragraph*{{Traditional formulation}}\label{ImageOfJTradtionalFormulation}

\hypertarget{description_of_the_image}{}\paragraph*{{Description of the image}}\label{description_of_the_image}

The following characterization of the [[image]] of the J-homomorphism on [[homotopy groups]] derives from a statement that was first conjectured in (\hyperlink{Adams66}{Adams 66}) -- and since called the \emph{[[Adams conjecture]]} -- and then proven in (\hyperlink{Quillen71}{Quillen 71}, \hyperlink{Sullivan74}{Sullivan 74}).

\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{\href{orthogonal%20group#HomotopyGroups}{orthogonal group -- homotopy groups}} we have that the [[homotopy groups]] of the [[stable orthogonal group]] are

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$n\;mod\; 8$&0&1&2&3&4&5&6&7\\
\hline 
$\pi_n(O)$&$\mathbb{Z}_2$&$\mathbb{Z}_2$&0&$\mathbb{Z}$&0&0&0&$\mathbb{Z}$\\
\end{tabular}

Because all groups appearing here and in the following are [[cyclic groups]], we instead write down the [[order of a group|order]]

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$n\;mod\; 8$&0&1&2&3&4&5&6&7\\
\hline 
${\vert\pi_n(O)\vert}$&2&2&1&$\infty$&1&1&1&$\infty$\\
\end{tabular}

\end{remark}
For the following statement it is convenient to restrict to J-homomorphism to the stable [[special orthogonal group]] $S O$, which removes the lowest degree homotopy group in the above

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$n\;mod\; 8$&0&1&2&3&4&5&6&7\\
\hline 
$\pi_n(S O)$&0&$\mathbb{Z}_2$&0&$\mathbb{Z}$&0&0&0&$\mathbb{Z}$\\
\end{tabular}

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$n\;mod\; 8$&0&1&2&3&4&5&6&7\\
\hline 
${\vert\pi_n(S O)\vert}$&1&2&1&$\infty$&1&1&1&$\infty$\\
\end{tabular}

\begin{theorem}
\label{AdamsQuillenTheorem}\hypertarget{AdamsQuillenTheorem}{}
The [[stable homotopy groups of spheres]] $\pi_n(\mathbb{S})$ are the [[direct sum]] of the ([[cyclic group|cyclic]]) [[image]] $im(J|_{SO})$ of the J-homomorphism, def. \ref{JHom}, applied to the [[special orthogonal group]] and the [[kernel]] of the [[Adams e-invariant]].

Moreover,

\begin{itemize}%
\item for $n = 0 \;mod \;8$ and $n = 1 \;mod \; 8$ and $n$ positive the J-homomorphism $\pi_n(J) \colon \pi_n(S O) \to \pi_n(\mathbb{S})$ is [[injection|injective]], hence its image is $\mathbb{Z}_2$,


\item for $n = 3\; mod\; 8$ and $n = 7 \; mod \; 8$ hence for $n = 4 k -1$, the [[order of a group|order]] of the image is equal to the [[denominator]] of $B_{2k}/4k$ in its reduced form, where $B_{2k}$ is the [[Bernoulli number]]


\item for all other cases the image is necessarily zero.



\end{itemize}
\end{theorem}
This characterization of the image of $J$ is due to (\hyperlink{Adams66}{Adams 66}, \hyperlink{Quillen71}{Quillen 71}, \hyperlink{Sullivan74}{Sullivan 74}). Specifically the identification of $J(\pi_{4n-1}(S O))$ is (\hyperlink{Adams65a}{Adams 65a, theorem 3.7} and the direct summand property is (\hyperlink{Adams66}{Adams 66, theorems 1.1-1.6.}). That the image is a direct summand of the codomain is proven for instance in (\hyperlink{Switzer75}{Switzer 75, end of chapter 19}).

A modern version of the proof, using methods from [[chromatic homotopy theory]], is surveyed in some detail in (\hyperlink{Lorman13}{Lorman 13}).

The statement of the theorem is recalled for instance as (\hyperlink{RavenelCh1}{Ravenel, chapter 1, theorem 1.1.13}). Another computation of the image of $J$ is in (\hyperlink{RavenelChapter5}{Ravenel, chapter 5, section 3}).

\begin{remark}
\label{}\hypertarget{}{}
The order of $J(\pi_{4k-1} O)$ in theorem \ref{AdamsQuillenTheorem} is for low $k$ given by the following table

\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l}
k&1&2&3&4&5&6&7&8&9&10\\
\hline 
$\vert J(\pi_{4k-1}(O))\vert$&24&240&504&480&264&65,520&24&16,320&28,728&13,200\\
\end{tabular}

\end{remark}
See for instance (\hyperlink{RavenelCh1}{Ravenel, Chapt. 1, p. 5}).

\begin{remark}
\label{}\hypertarget{}{}
Therefore we have in low degree the following situation

[[!include image of J -- table]]

\end{remark}
The following tables show the [[p-primary components]] of the [[stable homotopy groups of spheres]] for low values, the image of J appears as the bottom row.

Here the horizontal index is the degree $n$ of the stable homotopy group $\pi_n$. The appearance of a string of $k$ connected dots vertically above index $n$ means that there is a [[direct sum|direct summand]] [[primary group]] of [[order of a group|order]] $p^k$. See example \ref{InterpretTable} below for illustration.

(The tables are taken from (\hyperlink{Hatcher}{Hatcher}), where in turn were they were generated based on (\hyperlink{RavenelCh1}{Ravenel 86}).

\textbf{at $p = 2$}



\textbf{at $p = 3$}



\textbf{at $p = 5$}



\begin{example}
\label{InterpretTable}\hypertarget{InterpretTable}{}
The [[finite abelian group]] $\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_8 \oplus \mathbb{Z}_3$. Here $8 = 2^3$ corresponds to the three dots above $n = 3$ in the first table, and $3 = 3^1$ to the single dot over $n = 3$ in the second.

The [[finite abelian group]] $\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{24}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_{16} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5$. Here $16 = 2^4$ corresponds to the four dots above $n = 7$ in the first table, and $3 = 3^1$ to the single dot over $n = 7$ in the second and $5 = 5^1$ to the single dot over $n = 7$ in the third table.

\end{example}
\hypertarget{characterization_via_the_adams_operations}{}\paragraph*{{Characterization via the Adams operations}}\label{characterization_via_the_adams_operations}

(\ldots{})

We indicate how the [[Adams conjecture]]/Adams-Quillen-Sullivan theorem serves to identify the image of the J-homomorphism. We follow the modern account as reviewed in (\hyperlink{Lorman13}{Lorman 13}).

(\ldots{})

Write $\psi^k$ for the $k$th [[Adams operation]] on [[complex K-theory]].

Let $p$ be a [[prime]]. Consider $k$ coprime to $p$.

The [[Adams conjecture]] implies that completed at $p$, the J-homomorphism factors through the [[homotopy fiber]] of $1 - \psi^k$.

proof:

We have a homotopy-commuting diagram

\begin{displaymath}
\itexarray{
    B U_p &\stackrel{1 - \psi^k}{\longrightarrow}& B U_p
    \\
    \downarrow &\swArrow_\simeq& \downarrow
    \\
    \ast &\stackrel{0}{\longrightarrow}& B H_p
  }
  \,.
\end{displaymath}
The [[pasting]] composite with the [[homotopy pullback]] that witnesses the [[homotopy fiber]] of $1 - \psi^k$ induces via the [[universal property]] of the [[loop space object]] a canonical map $fib(1-\psi^k) \longrightarrow H_p$:

\begin{displaymath}
\itexarray{
    fib(1-\psi^k) &\longrightarrow& \ast
    \\
    \downarrow &&  \downarrow
    \\
    B U_p &\stackrel{1 - \psi^k}{\longrightarrow}& B U_p
    \\
    \downarrow &\swArrow_\simeq& \downarrow
    \\
    \ast &\stackrel{0}{\longrightarrow}& B H_p
  }
  \;\;\;
  \simeq 
  \;\;\;
  \itexarray{
    fib(1-\psi^k) &\longrightarrow& \ast
    \\
    \downarrow &&  \downarrow
    \\
    H_p &\stackrel{}{\longrightarrow}& \ast
    \\
    \downarrow &\swArrow_\simeq& \downarrow
    \\
    \ast &\stackrel{0}{\longrightarrow}& B H_p
  }   
  \,.
\end{displaymath}
(\ldots{})

\hypertarget{the_jspectrum}{}\paragraph*{{The J-spectrum}}\label{the_jspectrum}

The [[J-spectrum]] is a [[spectrum]] whose [[homotopy groups]] are close to being the image of the J-homomorphism.

(\ldots{})

\hypertarget{ImageOfJInChromotopy}{}\paragraph*{{Formulation in chromatic homotopy theory}}\label{ImageOfJInChromotopy}

In terms of [[chromatic homotopy theory]] the nature of the image of the J-homomorphism can be formulated more succinctly as follows.

Write $E(1)$ for the first [[Morava E-theory]] [[spectrum]] at given [[prime number]] $p$. Write $L_{E(1)}\mathbb{S}$ for the [[Bousfield localization of spectra]] of the [[sphere spectrum]] at $E(1)$.

\begin{theorem}
\label{}\hypertarget{}{}
The [[homotopy groups]] of the $E(1)$-localized sphere spectrum are

\begin{displaymath}
\pi_n L_{E(1)} \mathbb{S}
  \simeq
  \left\{
    \itexarray{
      \mathbb{Z} & if\; n = 0
       \\
       \mathbb{Q}_p/\mathbb{Z}_p & if\; n= -2
       \\
       \mathbb{Z}/p^{k+1}\mathbb{Z} & if\; n+1 = 2(p-1)p^k m \;with\; m \neq 0\;mod\;p
        \\
        0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
\end{theorem}
This appears as (\hyperlink{Lurie}{Lurie 10, theorem 6}) (note that there are two typos: first, the theorem is stated with $L_{K(1)} \mathbb{S}$ rather than $L_{E(1)} \mathbb{S}$, and second, it should be $n+1 = 2(p-1)p^k m$ but the $2$ is missing.)

\begin{defn}
\label{}\hypertarget{}{}
Write $\mathbb{S}_p$ for the [[p-localization]] of the [[sphere spectrum]]. For $n \in \mathbb{Z}$, write $im(J)_n$ for the [[image]] of the $p$-localized J-homomorphism

\begin{displaymath}
J \;\colon\; \pi_n(O) \longrightarrow \pi_n(\mathbb{S}) \longrightarrow \pi_n(\mathbb{S}_{(p)})
  \,.
\end{displaymath}
\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
For $n \in \mathbb{N}$, the further [[Bousfield localization]] at [[Morava E-theory|Morava E(1)-theory]] $\mathbb{S}_{(p)} \longrightarrow L_{E(1)}\mathbb{S}$ induces a [[isomorphism]]

\begin{displaymath}
im(J)_n \stackrel{\simeq}{\longrightarrow} \pi_n (L_{E(1)} \mathbb{S})
\end{displaymath}
between the image of the $J$-homomorphism and the $E(1)$-local [[stable homotopy groups of spheres]].

\end{theorem}
In this form this appears as (\hyperlink{Lurie}{Lurie 10, theorem 7}). See also (\hyperlink{Behrens13}{Behrens 13, section 1}).

\begin{cor}
\label{}\hypertarget{}{}
The $E(1)$-[[Bousfield localization of spectra|localization map]] is surjective on non-negative homotopy groups:

\begin{displaymath}
\pi_n(\mathbb{S}_{(p)}) \longrightarrow \pi_n(L_{E(1)} \mathbb{S})
  \,.
\end{displaymath}
\end{cor}
For review see also (\hyperlink{Lorman13}{Lorman 13}). That $J$ factors through $L_{K(1)}\mathbb{S}$ is in (\hyperlink{Lorman13}{Lorman 13, p. 4})

\begin{remark}
\label{}\hypertarget{}{}
Hence: the image of $J$ is essentially the first [[chromatic layer]] of the [[sphere spectrum]].

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

\begin{itemize}%
\item [[Thom spectra]], [[MO]]


\item [[orientation in generalized cohomology]]


\item [[J-homomorphism and chromatic homotopy]]


\item [[scanning map equivalence]]


\item [[twisted cohomotopy]]



\end{itemize}
$\backslash$linebreak

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

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

The J-homomorphism was introduced in

\begin{itemize}%
\item [[George Whitehead]], \emph{On the homotopy groups of spheres and rotation groups}, Annals of Mathematics. Second Series 43 (4): 634--640 (1942), (\href{http://www.jstor.org/stable/1968956}{JSTOR}).

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Akhil Mathew]], \emph{The Adams conjecture I} (\href{http://amathew.wordpress.com/2013/01/23/the-adams-conjecture-i/}{web})


\item [[Akhil Mathew]], \emph{Notes on the J-homomorphism}, 2012 (\href{http://math.uchicago.edu/~amathew/j.pdf}{pdf}, [[MathewJHomomorphism.pdf:file]])


\item Arpon Raksit, \emph{Vector fields and the J-homomorphism}, 2014 (\href{http://stanford.edu/~arpon/math/files/vfields.pdf}{pdf})



\end{itemize}
Discussion in [[higher algebra]] in term of [[(∞,1)-module bundles]] is in

\begin{itemize}%
\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], \emph{Twists of K-theory and TMF} (\href{http://arxiv.org/abs/1002.3004}{arXiv:1002.3004})

\end{itemize}
The [[complex geometry|complex]] J-homomorphism is discussed in

\begin{itemize}%
\item [[Victor Snaith]], \emph{The complex J-homomorphism}, Proc. London Math. Soc. (1977) s3-34 (2): 269-302 (\href{http://plms.oxfordjournals.org/content/s3-34/2/269.full.pdf+html}{journal})


\item [[Victor Snaith]], \emph{Infinite loop maps and the complex $J$-homomorphism}, Bull. Amer. Math. Soc. Volume 82, Number 3 (1976), 508-510. (\href{http://projecteuclid.org/euclid.bams/1183537922}{Euclid})



\end{itemize}
A [[p-adic number|p-adic]] J-homomorphism is described in

\begin{itemize}%
\item Dustin Clausen, \emph{p-adic J-homomorphisms and a product formula} (\href{http://arxiv.org/abs/1110.5851}{arXiv:1110.5851})

\end{itemize}
\hypertarget{the_image_of_j}{}\subsubsection*{{The image of J}}\label{the_image_of_j}

The analysis of the image of $J$ is due to

\begin{itemize}%
\item [[John Adams]], \emph{On the groups $J(X)$ I}, Topology 2 (3) (1963) (\href{http://math1.unice.fr/~cazanave/Gdt/ImJ/J-I.pdf}{pdf})


\item [[John Adams]], \emph{On the groups $J(X)$ II}, Topology 3 (2) (1965) (\href{http://math1.unice.fr/~cazanave/Gdt/ImJ/J-II.pdf}{pdf})


\item [[John Adams]], \emph{On the groups $J(X)$ III}, Topology 3 (3) (1965) (\href{http://math1.unice.fr/~cazanave/Gdt/ImJ/J-III.pdf}{pdf})


\item [[John Adams]], \emph{On the groups $J(X)$ IV}, Topology 5: 21,(1966) \emph{Correction}, Topology 7 (3): 331 (1968) (\href{http://math.unice.fr/~cazanave/Gdt/ImJ/J-IV.pdf}{pdf})


\item [[Daniel Quillen]], \emph{The Adams conjecture}, Topology. an International Journal of Mathematics 10: 67--80 (1971) (\href{http://math1.unice.fr/~cazanave/Gdt/ImJ/Quillen.pdf}{pdf})


\item [[Dennis Sullivan]], \emph{Genetics of homotopy theory and the Adams conjecture}, Ann. of Math. 100 (1974), 1--79.


\item [[Robert Switzer]], \emph{Algebraic topology--homotopy and homology}, Springer-Verlag, New York, 1975.



\end{itemize}
The statement of the theorem about the characterization of the image is reviewed in

\begin{itemize}%
\item [[Doug Ravenel]], \emph{[[Complex cobordism and stable homotopy groups of spheres]]}, chapter 1, \emph{An introduction to the homotopy groups of spheres} (\href{http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel1.pdf}{pdf})

\end{itemize}
see there also around theorem 3.4.16.

The details of the proof are surveyed in

\begin{itemize}%
\item [[Doug Ravenel]], \emph{[[Complex cobordism and stable homotopy groups of spheres]]}, chapter 5, \emph{The chromatic spectral sequence} (\href{http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel5.pdf}{pdf})

\end{itemize}
Tables showing the image of $J$ at low primes are in

\begin{itemize}%
\item [[Allen Hatcher]], \emph{Stable homotopy groups of spheres} (\href{http://www.math.cornell.edu/~hatcher/stemfigs/stems.html}{html})

\end{itemize}
Other reviews include

\begin{itemize}%
\item [[Mark Mahowald]], \emph{The Image of J in the EHP Sequence}, Annals of Mathematics Second Series, Vol. 116, No. (\href{http://www.jstor.org/stable/2007048}{JSTOR})


\item [[Johannes Ebert]], \emph{The Adams conjecture after Edgar Brown}, (\href{http://www.math.uni-muenster.de/u/jeber_02/talks/adams.pdf}{pdf})



\end{itemize}
Discussion from the point of view of [[chromatic homotopy theory]] is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{The image of $J$}, lecture 35 in \emph{[[Chromatic Homotopy Theory]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture35.pdf}{pdf})


\item [[Mark Behrens]], section 1 of Introduction talk at \emph{\href{http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks}{Talbot 2013: Chromatic Homotopy Theory}} (\href{http://math.mit.edu/conferences/talbot/2013/1-Behrens-intro.pdf}{pdf}, \href{http://math.mit.edu/conferences/talbot/2013/Behrens-Introduction-chromatic-homotopy-thy-4-22-13.pdf}{pdf})


\item [[Ben Knudsen]], \emph{First chromatic layer of the sphere spectrum = homotopy of the $K(1)$-local sphere}, talk at \emph{\href{http://math.northwestern.edu/~htanaka/pretalbot2013/index.php}{2013 Pre-Talbot Seminar}} (\href{http://math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-04-04-Rob-Legg-K1-local-sphere.pdf}{pdf})


\item [[Vitaly Lorman]], \emph{Chromatic homotopy theory at height 1 and the image of $J$}, talk at \emph{\href{http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks}{Talbot 2013: Chromatic Homotopy Theory}} (\href{http://math.mit.edu/conferences/talbot/2013/Image%20of%20J-1.pdf}{pdf})



\end{itemize}
Generalization to [[equivariant cohomology]] ([[equivariant K-theory]]) is discussed in

\begin{itemize}%
\item Z. Fiedorowicz, H. Hauschild, [[Peter May]], theorem 0.4 of \emph{Equivariant algebraic K-theory}, \emph{Equivariant algebraic K-theory}, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 (\href{http://math.uchicago.edu/~may/PAPERS/40.pdf}{pdf})


\item Henning Hauschild, [[Stefan Waner]], theorem 0.1 of \emph{The equivariant Dold theorem mod $k$ and the Adams conjecture}, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. (\href{https://projecteuclid.org/euclid.ijm/1256065410}{euclid:1256065410})


\item Kuzuhisa Shimakawa, \emph{Note on the equivariant $K$-theory spectrum}, Publ. RIMS, Kyoto Univ. \textbf{29} (1993), 449-453 (\href{http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=29&iss=3&rank=5}{pdf}, \href{https://doi.org/10.2977/prims/1195167052}{doi})


\item Christopher French, theorem 2.4 in \emph{The equivariant $J$–homomorphism for finite groups at certain primes}, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 (\href{https://projecteuclid.org/euclid.agt/1513797069}{euclid:1513797069})



\end{itemize}
\hypertarget{equivariant_enhancement}{}\subsubsection*{{Equivariant enhancement}}\label{equivariant_enhancement}

The generalization to [[equivariant cohomology]] ([[equivariant K-theory]] and the [[equivariant Adams conjecture]]) is discussed in

\begin{itemize}%
\item [[Tammo tom Dieck]], theorem 11.3.8 in \emph{[[Transformation Groups and Representation Theory]]} Lecture Notes in Mathematics 766 Springer 1979


\item Z. Fiedorowicz, H. Hauschild, [[Peter May]], theorem 0.4 of \emph{Equivariant algebraic K-theory}, \emph{Equivariant algebraic K-theory}, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 (\href{http://math.uchicago.edu/~may/PAPERS/40.pdf}{pdf})


\item Henning Hauschild, [[Stefan Waner]], theorem 0.1 of \emph{The equivariant Dold theorem mod $k$ and the Adams conjecture}, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. (\href{https://projecteuclid.org/euclid.ijm/1256065410}{euclid:1256065410})


\item Kuzuhisa Shimakawa, \emph{Note on the equivariant $K$-theory spectrum}, Publ. RIMS, Kyoto Univ. \textbf{29} (1993), 449-453 (\href{http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=29&iss=3&rank=5}{pdf}, \href{https://doi.org/10.2977/prims/1195167052}{doi})


\item Christopher French, theorem 2.4 in \emph{The equivariant $J$–homomorphism for finite groups at certain primes}, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 (\href{https://projecteuclid.org/euclid.agt/1513797069}{euclid:1513797069})



\end{itemize}
\hypertarget{relation_to_action_on_general_spectra}{}\subsubsection*{{Relation to $O$-action on general spectra}}\label{relation_to_action_on_general_spectra}

Similarly there is a canonical $O(n)$-[[∞-action]] on an [[n-fold loop space]], not just on the [[sphere spectrum]]. But the general case is closely related to the J-homomorphism. Discussion includes

\begin{itemize}%
\item Gerald Gaudens, [[Luc Menichi]], section 5 of \emph{Batalin-Vilkovisky algebras and the $J$-homomorphism}, Topology and its Applications Volume 156, Issue 2, 1 December 2008, Pages 365--374 (\href{http://arxiv.org/abs/0707.3103}{arXiv:0707.3103})

\end{itemize}
and in the context of the [[cobordism hypothesis]]:

\begin{itemize}%
\item [[Jacob Lurie]], example 2.4.15 of \emph{[[On the Classification of Topological Field Theories]]}, Current Developments in Mathematics Volume 2008 (2009), 129-280 (\href{http://arxiv.org/abs/0905.0465}{arXiv:0905.0465})

\end{itemize}
[[!redirects J-homomorphisms]]

[[!redirects image of J]] [[!redirects image of J-homomorphism]] [[!redirects image of the J-homomorphism]]

[[!redirects equivariant J-homomorphism]] [[!redirects equivariant J-homomorphisms]]



\end{document}