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\section*{classical Adams spectral sequence}

\begin{quote}%
This entry is about the classical Adams spectral sequence only. For more general discussion see at \emph{[[Adams spectral sequence]]}.

under construction


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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{MotivationFromHurewiczTheoremAndSerreSpectralSequence}{Via iterated Hurewicz theorem and Serre spectral sequence}\dotfill \pageref*{MotivationFromHurewiczTheoremAndSerreSpectralSequence} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{generators_in_low_range}{Generators in low range}\dotfill \pageref*{generators_in_low_range} \linebreak
\noindent\hyperlink{further}{Further}\dotfill \pageref*{further} \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{classical Adams spectral sequence} (\hyperlink{Adams58}{Adams 58}) is a type of [[spectral sequences]] used for computations in [[stable homotopy theory]]. It computes the [[homotopy groups of spheres]] at prime 2 from [[homology]]/[[cohomology]], as [[modules]]/[[comodules]] over its [[Steenrod operations]]. The Adams spectral sequence may be seen as a variant of the [[Serre spectral sequence]] obtained by replacing a single fibration by an ``[[Adams resolution]]''.

The original \emph{clasical Adams spectral sequenc} for [[ordinary cohomology]] is further refined by the \emph{[[Adams-Novikov spectral sequence]]} (\hyperlink{Novikov67}{Novikov 67}) by replacing [[ordinary cohomology]] modulo $p$ by [[complex cobordism cohomology theory]] or [[Brown-Peterson theory]] or the like.

Generally, for $E$ a suitable [[E-infinity algebra]] there is a corresponding \emph{$E$-[[Adams spectral sequence]]} whose second page is given by $E$-[[generalized cohomology]] and which arises as the [[spectral sequence of a simplicial stable homotopy type]] of the [[cosimplicial object|cosimplicial]] object which is the [[Cech nerve]]/[[Sweedler coring]]/[[Amitsur complex]] of $E$. As such the Adams spectral sequence is an analog in [[stable homotopy theory]] of the [[Bousfield-Kan spectral sequence|Bousfield-Kan]] [[homotopy spectral sequence]] in unstable [[homotopy theory]].

\hypertarget{MotivationFromHurewiczTheoremAndSerreSpectralSequence}{}\subsubsection*{{Via iterated Hurewicz theorem and Serre spectral sequence}}\label{MotivationFromHurewiczTheoremAndSerreSpectralSequence}

The classical Adams spectral sequence may be motivated from the strategy to compute [[homotopy groups]] from [[cohomology groups]] by subsequently applying the [[Hurewicz theorem]] to compute the lowest-degree non-trivial homotopy group from the corresponding [[cohomology group]], then co-killing that by forming its [[homotopy fiber]], finally applying the [[Serre spectral sequence]] to identify the next lowest non-trivial cohomology group of that fiber, and then iterating this process. The Adams spectral sequence arises when in this kind of strategy instead of co-killing only the lowest lying [[cohomology group]], one at a time, one co-kills \emph{all} nontrivial cohomology groups, then forms the corresponding [[homotopy fiber]] and so on.

This was apparently historically the way that [[John Adams]] indeed proceeded from [[Jean-Pierre Serre]]`s approach and this is still a good motivation for the whole construction. A nice exposition is in (\hyperlink{Wilson13}{Wilson 13, 1.1}).

We now say this again in more detail.

Given $n \in \mathbb{N}$, consider the probem of computing the [[homotopy groups of spheres|homotopy groups]] $\pi_k(S^n) \;mod \;2$ of the [[n-sphere]] $S^n$. For $k \leq n$ this is clear: first for $k \lt n$ they all vanish, and second for $k = n$ we have, by the very nature of [[Eilenberg-MacLane spaces]] $K(\mathbb{Z}_2, n)$, that the [[ordinary cohomology]] is

\begin{displaymath}
H^n(S^n, \mathbb{Z}_2) \simeq [S^n, K(\mathbb{Z}_2,n)] \simeq \pi_n(K(\mathbb{Z}_2,n)) \simeq \mathbb{Z}_2
\end{displaymath}
so that by the [[Hurewicz theorem]] it follows that also

\begin{displaymath}
\pi_n(S^n) \;mod\;2 \;\simeq \mathbb{Z}_2
  \,.
\end{displaymath}
The [[Hurewicz theorem]] does not say anything beyond the first non-vanishing cohomology group, but so to apply it again we can move up one step in the [[Whitehead tower]] of $S^n$ and hence consider the [[homotopy fiber]]

\begin{displaymath}
\itexarray{
     F_1
     \\
     \downarrow
     \\
     S^n &\stackrel{c_1}{\longrightarrow}& K(\mathbb{Z}_2,n)
  }
\end{displaymath}
of the generator $[c_1] = 1 \in \pi_n(S^n) \simeq \mathbb{Z}_2$.

To apply the [[Hurewicz theorem]] to that fiber we need to know its lowest non-trivial [[cohomology group]] again, and this is computed via the [[Serre spectral sequence]] applied to this [[fiber sequence]].

From here on the process repeats, and one moves higher through the [[Whitehead tower]] of $S^n$

\begin{displaymath}
\itexarray{
     \vdots
     \\
     \downarrow
     \\
     F_1 &\stackrel{c_2}{\longrightarrow}& K(\mathbb{Z}_2, n+1)
     \\
     \downarrow
     \\
     S^n &\stackrel{c_1}{\longrightarrow}& K(\mathbb{Z}_2,n)
  }
  \,.
\end{displaymath}
The Adams spectral sequence arises from this strategy by co-killing not just the first non-trivial [[cohomology group]] at each stage, but \emph{all} nontrivial cohomology groups at a given stage.

This is done in [[stable homotopy theory]], so let now $X$ be a [[spectrum]] (for instance the [[sphere spectrum]] $X = \mathbb{S}$ if we still with the computation of the [[stable homotopy groups of spheres]]). Write $H \mathbb{F}_2$ for the [[Eilenberg-MacLane spectrum]] for [[ordinary cohomology]] with [[coefficients]] in $\mathbb{Z}_2$, so that an element in [[cohomology]]

\begin{displaymath}
[c] \in H^n(X)
\end{displaymath}
is represented by the [[homotopy class]] of a [[homomorphism]] of [[spectra]] of the form

\begin{displaymath}
c \;\colon\;  X \longrightarrow \Sigma^n H\mathbb{F}_2
\end{displaymath}
(a [[cocycle]]), where ``$\Sigma$'' denotes [[suspension]], as usual.

If $X$ is a [[spectrum]] of [[finite type]] then there is a [[finite]] $I$ of non-trivial cohomology classes like this, and a choice of [[cocycles]] $c_i$ for each of them gives a single map

\begin{displaymath}
f_0 
    \coloneqq 
   (c_i)_I \;\colon\; 
   X \longrightarrow K_0 \coloneqq \bigvee_{i \in I} \Sigma^{n_i}H \mathbb{F}_2
\end{displaymath}
into a [[generalized Eilenberg-MacLane spectrum]]. As before, this map classifies its [[homotopy fiber]]

\begin{displaymath}
\itexarray{
    F_1
    \\
    \downarrow
    \\
     X &\stackrel{f_0}{\longrightarrow}& K_0
  }
\end{displaymath}
which may be thought of as encoding all information about $X$ beyond its [[cohomology groups]]. Iterating this process gives the corresponding analog of the [[Whitehead tower]], called the \emph{[[Adams resolution]]} of $X$:

\begin{displaymath}
\itexarray{
    \vdots
    \\
    \downarrow
    \\
    F_2 &\stackrel{f_2}{\longrightarrow}& K_2
    \\
    \downarrow
    \\
    F_1 &\stackrel{f_1}{\longrightarrow}& K_1
    \\
    \downarrow
    \\
     X &\stackrel{f_0}{\longrightarrow}& K_0
  }
  \,.
\end{displaymath}
The \emph{Adams spectral sequence} is that induced by the [[exact couple]] obtained by applying $\pi_\bullet$ to this [[Adams resolution]].

We now say this more in detail.

The [[long exact sequences of homotopy groups]] for all the [[homotopy fibers]] in this diagram arrange into a diagram of the form

\begin{displaymath}
\itexarray{
    \vdots
    \\
    \downarrow & \nwarrow
    \\
    \pi_\bullet(F_2) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}& \pi_\bullet(K_2)
    \\
    \downarrow & \nwarrow^{\mathrlap{\partial_2}}
    \\
    \pi_\bullet(F_1) &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}& \pi_\bullet(K_1)
    \\
    \downarrow & \nwarrow^{\mathrlap{\partial_1}}
    \\
    \pi_\bullet(X) &\stackrel{\pi_\bullet(f_0)}{\longrightarrow}& \pi_\bullet(K_0)
  }
  \,,
\end{displaymath}
where the diagonal maps are the [[connecting homomorphisms]] and hence decrease degree in $\pi_\bullet$ by one. The idea now is to compute the [[homotopy groups]] of $X$ from the decomposed information in this diagram as follows.

First, by construction the homotopy groups $\pi_\bullet(K_s)$ are known, therefore we can identify elements

\begin{displaymath}
\sigma \in \pi_\bullet(X)
\end{displaymath}
if they come from elements

\begin{displaymath}
\sigma_s \in \pi_\bullet(X_s)
\end{displaymath}
whose image

\begin{displaymath}
\pi_\bullet(f_s)(\sigma_s) \in \pi_\bullet(K_s)
\end{displaymath}
we understand. So the task is to understand the image of $\pi_\bullet(f_s)$ in $\pi_\bullet(K_s)$, for each $s$.

By [[exact sequence|exactness]] an element $\kappa_s \in \pi_\bullet(K_s)$ is in this image if its image

\begin{displaymath}
\rho_{s+1} \coloneqq \partial(\kappa_s) \in \pi_{\bullet-1}(X_{s+1})
\end{displaymath}
vanishes. Now, by construction of the resolution, ``evidence'' for this is that $f_{s+1}(\partial(\kappa_s)) \in \pi_{\bullet-1}(K_{s+1})$ vanishes, which in turn by [[exact sequence|exactness]] means equivalently that $\partial(\kappa_s)$ is the image of an element $\rho_{s+2} \in  \pi_{\bullet-1}(X_{s+2}) \to \pi_{\bullet-1}(X_{s+1})$. Now again ``evidence'' for $\rho_{s+2}$ to vanish is that its image $f_{s+2}(\rho(s+2))$ vanishes, which again means that it comes from an element $\rho_{s+3} \in  \pi_{\bullet-1}(X_{s+3}) \to \pi_{\bullet-1}(X_{s+2})$.

Proceeding by [[induction]] this way, we find that accumulated ``evidence'' in homotopy groups of $K_\bullet$ for an element $\kappa_s$ to represent an element in $\pi_\bullet(X)$ is that its differential $\partial \kappa_s$ factors through all the $\pi_{\bullet-1}(X_{s+k}) \to \pi_{\bullet-1}(X_s)$. This in turn means that it factors through the [[inverse limit]] $\underset{\leftarrow}{\lim}_s \pi_{\bullet-1}(X_s)$. Such an element $\kappa_s$ with

\begin{displaymath}
\partial \kappa_s \in \underset{\leftarrow}{\lim}_s \pi_{\bullet-1}(X_s) \to \pi_{\bullet-1}(X_{s+1})
\end{displaymath}
is called a \emph{permanent cycle}.

In good cases, the [[Adams resolution]] is indeed a [[resolution]] which means that the [[inverse limit]] $\underset{\leftarrow}{\lim}_s X_s$ is in fact [[contractible]]. This means that all the ``evidence'' accumulated in a permanent cycle is indeed sufficient evidence to prove the existence of an element $\sigma_s \in \pi_\bullet(X_s)$ and hence of an element $\sigma \in \pi_\bullet(X)$.

A trivial way for this to be the case is that the original $\sigma_s$ is itself in the image under $\partial$ of some element, in which case $\kappa_s = 0$ already all by itself. These elements are called \emph{eventual boundaries}. Therefore if the Adams resolution is indeed a resolution, then the quotient group

\begin{displaymath}
\frac{permanent\;cycles}{eventual\;boundaries}
\end{displaymath}
gives elements in $\pi_\bullet(S)$, and this quotient is what the Adams spectral sequence computes.

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

\hypertarget{generators_in_low_range}{}\subsubsection*{{Generators in low range}}\label{generators_in_low_range}

For the moment see at \emph{[[May spectral sequence]]}.

\hypertarget{further}{}\subsubsection*{{Further}}\label{further}

$h_j$ is a permanent cycle in the Adams spectral sequence if $\mathbb{R}^{(2^j)}$ admits the structure of a real [[division algebra]]

Adams: \ldots{}

\begin{displaymath}
d_2(h_{j+1}) = h_0 h_j^2
\end{displaymath}
Mahowald-Tangora: $h_4^2$ is a permanent cycle

Barratt-Jones-Mahowald: $h_5^2$ is a permanent cycle

Hill-Hopkins-Ravenel: for $j \gt 7$ then $h_j^2$ is not a permanent cycle.

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

\begin{itemize}%
\item [[May spectral sequence]], [[Curtis algorithm]]


\item [[Adams-Novikov spectral sequence]]



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

The original article is

\begin{itemize}%
\item [[Frank Adams]], \emph{On the structure and applications of the Steenrod algebra}, Comm. Math. Helv. 32 (1958), 180--214.

\end{itemize}
Textbook accounts proceeding in the [[coalgebra]] picture include

\begin{itemize}%
\item [[Doug Ravenel]], chapter 3 of \emph{[[Complex cobordism and stable homotopy groups of spheres]]}, 1986/2003


\item [[Stanley Kochman]], chapter 5 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996



\end{itemize}
Further review includes

\begin{itemize}%
\item [[Robert Mosher]], [[Martin Tangora]], section 18 of \emph{Cohomology Operations and Application in Homotopy Theory}, Harper and Row (1968) (\href{www.maths.ed.ac.uk/~aar/papers/moshtang.pdf}{pdf})


\item [[John McCleary]], chapter 9 of \emph{A User's Guide to Spectral Sequences}, Cambridge University Press (1985, 2001)


\item [[Robert Bruner]], \emph{An Adams spectral sequence primer}, 2009 (\href{http://www.math.wayne.edu/~rrb/papers/adams.pdf}{pdf})


\item [[John Rognes]], \emph{The Adams spectral sequence} (following \hyperlink{Bruner09}{Bruner 09}), 2012 (\href{http://folk.uio.no/rognes/papers/notes.050612.pdf}{pdf})



\end{itemize}
and

\begin{itemize}%
\item [[Alexander Kupers]], \emph{An introduction to the Adams spectral sequence} (following \hyperlink{Rognes12}{Rognes 12}) (\href{http://math.stanford.edu/~kupers/adamsss.pdf}{pdf})


\item Paolo Masulli, \emph{Stable homotopy and the Adams spectral sequence} (\href{http://www.math.ku.dk/~jg/students/masulli.msproject.2011.pdf}{pdf})


\item \href{http://www.math.harvard.edu/~sia/notes/classical_topology_adams.pdf}{pdf}



\end{itemize}


\end{document}