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\begin{document}

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\section*{EHP spectral sequence}

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

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

[[!include homotopy - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{constructions_of_}{Constructions of $H$}\dotfill \pageref*{constructions_of_} \linebreak
\noindent\hyperlink{via_the_james_model}{Via the James model}\dotfill \pageref*{via_the_james_model} \linebreak
\noindent\hyperlink{via_pushoutpullback_comparisons}{Via pushout/pullback comparisons}\dotfill \pageref*{via_pushoutpullback_comparisons} \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 \emph{EHP spectral sequence} (we follow \hyperlink{Mahowald85}{Mahowald 85}) is the [[spectral sequence]] for computation of [[homotopy groups of spheres]] induced from the [[filtration]] of the underlying homotopy type $\Omega^\infty \Sigma^\infty S^0 = \Omega^\infty \mathbb{S}$ of the [[sphere spectrum]] by [[suspensions]] (German: \emph{E}inh\"a{}ngung):

\begin{displaymath}
\Omega^n S^n \stackrel{E}{\longrightarrow} \Omega^{n+1} S^{n+1}
  \,.
\end{displaymath}
More concretely, (\hyperlink{James57}{James 57}) constructed maps

\begin{displaymath}
\Omega S^n \stackrel{H}{\longrightarrow} \Omega S^{2n-1}
\end{displaymath}
(for \emph{H}opf as in [[Hopf invariant]]) and showed that [[p-localization|2-locally]] these fit with $E$ into [[homotopy fiber sequences]]

\begin{displaymath}
\Omega^{n+2} S^{2n+1} 
    \stackrel{P}{\longrightarrow}
  \Omega^n S^n 
    \stackrel{E}{\longrightarrow} 
  \Omega^{n+1} S^{n+1}
    \stackrel{H}{\longrightarrow}
  \Omega^{n+1}S^{2n+1}
  \,.
\end{displaymath}
(Here $P$ is by definition the [[homotopy fiber]] of $E$, the notation refers to \emph{[[Whitehead product]]}.)

This ``EHP-[[long homotopy fiber sequence]]'' gives rise to the corresponding [[long exact sequence of homotopy groups]] and so to an [[exact couple]] of the form

\begin{displaymath}
\itexarray{
     \underset{s,t}{\oplus} \pi_{s+t}(\Omega^{s+1}S^{s+1})
     && \stackrel{i}{\longrightarrow} &&
     \pi_{s+t}(\Omega^{s+1}S^{s+1})
     \\
     & \nwarrow && \swarrow
     \\
     &&
     \underset{s,t}{\oplus} 
     \pi_{t+s}(\Omega^{s+1}S^{2s+1})
  }
  \,.
\end{displaymath}
The corresponding [[spectral sequence]] is the EHP spectral sequence proper. It converges, 2-locally, to the [[stable homotopy groups of spheres]], with $E_1$-page given by

\begin{displaymath}
E^{k,n}_1 = \pi_{k+n}(S^{2n-1}) \;\Rightarrow\; \pi_\bullet^{\mathbb{S}}
  \,.
\end{displaymath}
For more general [[prime numbers]] than just 2, (\hyperlink{Toda62}{Toda 62}) found analogous fibrations, which hence give EHP spectral sequences for general $p$.

The EHP spectral sequence is often used used in the context of the [[Adams-Novikov spectral sequence]] for [[p-localization]] at some prime $p$.

\hypertarget{constructions_of_}{}\subsection*{{Constructions of $H$}}\label{constructions_of_}

For James' fiber sequence, the essential property required of $H$ is to realize the isomorphism

\begin{displaymath}
(the\:map\:H)^{cohomology\:pullback} : H^{2n}(\Omega\mathbb{S}^{2n+1})\overset{\sim}{\to} H^{2n}(\Omega \mathbb{S}^{n+1}) .
\end{displaymath}
The remaining corollaries then follow using the fact cohomology pullback is a ring homomorphism, and the $mod 2$ [[Leray-Serre spectral sequence]].

\hypertarget{via_the_james_model}{}\subsubsection*{{Via the James model}}\label{via_the_james_model}

Using the James model of $\Omega\Sigma X$ as a quotient space of $colim_n X^n$, a candidate $H$ is constructed by recursion:

\begin{displaymath}
H([x_0 , \dots, x_{n+1}]) = H([x_0,\dots,x_n]) \# ([x_0 \wedge x_{n+1} , \dots , x_n\wedge x_{n+1}])
\end{displaymath}
where $\#$ denotes concatenation and $\wedge$ [[smash product]]. One checks that the ordering of product terms $x_i\wedge x_j$ w.r.t. $x_k\wedge x_l$ depends only on the relative orders of $i,j,k,l$, so that $H$ is well-defined on the quotient space $\Omega\Sigma X \to \Omega \Sigma(X\wedge X)$.

In particular, the restriction to $J_2 X$ factors through $X\wedge X \to \Omega\Sigma X\wedge X$ as the cofiber of the inclusion $X \to J_2 X$. In the case $X\simeq \mathbb{S}^n$, the desired cohomology isomorphism is immediate.

\hypertarget{via_pushoutpullback_comparisons}{}\subsubsection*{{Via pushout/pullback comparisons}}\label{via_pushoutpullback_comparisons}

Starting with the three-legged cospan $X \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} *$, construct the cube of all pushouts

\begin{displaymath}
\itexarray{
   X    &    &              & \to &       &     &    * \\
    & \searrow &             &   &     & \swarrow \\
            &       &   *   & \to &   \Sigma X \\
  \downarrow & & \downarrow &     & \downarrow & & \downarrow \\
            &    & \Sigma X & \to & \Sigma X \vee \Sigma X \\
         & \nearrow  & &     &   &       \nwarrow \\
   *       &      &         & \to &       &      & \Sigma X 
 }
\end{displaymath}
Construct pullbacks in some pair of parallel squares, and compare them by naturality

\begin{displaymath}
\itexarray{
   \Omega\Sigma X &   &           & \to &           &           & * \\
        & \searrow &           &     &           & \swarrow \\
        &   & \Omega((\Omega\Sigma X)\star(\Omega\Sigma X)) & \to & \Sigma X \\
   \downarrow   &    & \downarrow &     & \downarrow &        & \downarrow \\
             &       & \Sigma X   & \to & \Sigma X \vee \Sigma X \\
             & \nearrow &         &     &               & \nwarrow \\
       *     &          &         & \to &            &          & \Sigma X 
 }
\end{displaymath}
where $\star$ is [[join|reduced join]]. On the other hand, the natural transformations $\Sigma\Omega \to 1$ give natural maps, e.g.

\begin{displaymath}
\Omega( ev \circ \tau \circ ev ) : \Omega((\Omega\Sigma X) \star (\Omega\Sigma X)) \to \Omega( X \star X) .
\end{displaymath}
The composite

\begin{displaymath}
\Omega \Sigma X \to \Omega((\Omega\Sigma X)\star(\Omega\Sigma X)) \to \Omega(X\star X)
\end{displaymath}
is a candidate $H$.

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

\begin{itemize}%
\item the [[Freudenthal suspension theorem]] may be obtained from the EHP spectral sequence;


\item the [[Goodwillie spectral sequence]] of the identity functor at the point also computes homotopy groups of spheres, the interplay of the two is discussed in (\hyperlink{Behrens10}{Behrems 10})



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

Original articles include

\begin{itemize}%
\item [[Ioan Mackenzie James]], \emph{Reduced product spaces}, Ann. of Math. (2) 62 (1955), 170-197.


\item [[Ioan Mackenzie James]], \emph{On the Suspension Sequence}, Annals of Mathematics Second Series, Vol. 65, No. 1 (Jan., 1957), pp. 74-107 (\href{http://www.jstor.org/stable/1969666}{jstor})


\item [[Hiroshi Toda]], \emph{Composition methods in homotopy groups of spheres}, Princeton University Press (1962)


\item [[Mark Mahowald]], \emph{Lin's theorem and the EHP sequence}. Conference on algebraic topology in honor of [[Peter Hilton]], Contemp. Math. 37 (1985), 115--119. Amer. Math. Soc., Providence, RI.


\item [[Marcel Bökstedt]], Anne Marie Svane, \emph{A generalization of the stable EHP spectral sequence} (\href{http://arxiv.org/abs/1208.3938}{arXiv:1208.3938})



\end{itemize}
Relation to the [[Goodwillie spectral sequence]] is discussed in

\begin{itemize}%
\item [[Mark Behrens]], \emph{The Goodwillie tower and the EHP sequence} (\href{http://arxiv.org/abs/1009.1125}{arXiv:1009.1125})

\end{itemize}
An algebraic version of the EHP spectral sequence for the [[Lambda-algebra]] and used for computation of the second page of the [[classical Adams spectral sequence]] (the \emph{[[Curtis algorithm]]}), is discussed in

\begin{itemize}%
\item [[Stanley Kochman]], around prop. 5.2.6 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996

\end{itemize}
Review includes

\begin{itemize}%
\item [[Mark Mahowald]], [[Doug Ravenel]], section 7 of \emph{Towards a Global Understanding of the Homotopy Groups of Spheres} (\href{http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf}{pdf})


\item [[Doug Ravenel]], chapter 1, section 5 of \emph{[[Complex cobordism and stable homotopy groups of spheres]]}


\item [[eom]], \emph{\href{https://www.encyclopediaofmath.org/index.php/EHP_spectral_sequence}{EHP spectral sequence}}


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/EHP_spectral_sequence}{EHP spectral sequence}}



\end{itemize}


\end{document}