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\begin{document}

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{\Large\sf 
Documentation of the minimal resolution \\
\normalsize
($h_i$ multiplications and minimal attachment) \\
\bigskip
sourcefiles: \tt ahss.tex, ahssdat.tex
}

\bigskip

{\footnotesize
\tt\footnotesize
Christian Nassau \\
FB Mathematik (12) Ag 8:1\\
Johann Wolfgang Goethe Universit\"at \\
Robert Mayer Strasse 6-8 \\
60054 Frankfurt \\
Germany 

e-mail: nassau@math.uni-frankfurt.de
}

\vfill

{\bf Introduction.} \\
This document tries to give a documentation of the classical Adams E$_2$ term,
the information being extracted from (the analysis of) 
a minimal resolution of the mod 2 Steenrod algebra $A$.
It contains complete information about $h_i$ divisibility, and gives for each
generator of the resolution its {\em minimal attachment}. 
This last notion should be explained:

The resolution that has been computed comes equipped with a chosen basis
of $A$ generators $g_i\in C_s$ for each $s$.
Each generator $g\in C_s$ is attached to the generators $g'_i\in C_{s-1}$
in the sense that 
\begin{equation}\label{eq}
dg = \sum_{i} \sum_{j\in K_i} a_{i,j} g'_i
\end{equation}
for certain Milnor basis elements $a_{i,j}$. 
The minimal attachment is defined to be the sum of those $a_{i,j} g'_i$ for 
which the degree of $a_{i,j}$ is minimal. 
Note that the expressions (1) depend on the choice of the generators,
and are not invariants of the resolution.

The $h_i$ divisibility is computed from (1)  
via
$$ h_i g' \owns g \quad \Longleftrightarrow \quad {\rm Sq}^{2^i} g' \in dg,$$
where `$\in$' means `is a summand of'. 

The minimal attachment is included in this description because it seems to
give information on those generators which are not $h_i$ multiples of lower
classes, and thus might otherwise have no description at all in this file.
I believe the minimal attachment is related to the differentials in the
Atiyah-Hirzebruch spectral sequence
$$
A_\ast \otimes {\rm Ext}_A(F_2,F_2) \Rightarrow {\rm Ext}_A(A,F_2).
$$
The verification of this claim should be a straightforward exercise 
but so far I have been too lazy to work this out in detail. % :)
To give an example: if the generator $g$ has minimal attachment
$({\rm Sq(3) + Sq(0,1)}) g'$ this seems to say that in the AHSS $g$ is 
defined (= killed) by the differential 
$g = d\left(\left(\zeta_1^3 + \zeta_2\right)\otimes g'\right)$
where  $\zeta_i$ is the conjugate of the usual $\xi_i$. 
In other words we would have $g=d(\xi_1^3 \otimes g')$ which by
standard formulas\footnote{%
see Kochman's SLNM book (vol.\ 1423) for prototypes of such formulas, 
for example Thm.\ 2.4.4.} 
means something like
$g = \langle h_0, h_1, g'\rangle$ modulo terms of lower AHSS filtration
($h_0$ and $h_1$ multiples in this case).

In some bidegrees the minimal attachment is too complicated to be given
entirely. In that case only part of the attachment is given.  

\newpage

\thispagestyle{empty}

{\bf How to read this table.} \\
The following are some typical entries:
\begin{center}
% example 
\begin{bdl}
\item[10/40]
\begin{gl}
\item[13]
{\rm
Sq(2)[18]}
\\ $h_{1}:$   [18]
\end{gl}
\end{bdl}

\begin{bdl}
\item[8/40]
\begin{gl}
\item[18]
{\rm
Sq(0,2)[12]}
\end{gl}
\end{bdl}

\begin{bdl}
\item[6/40]
\begin{gl}
\item[18]
{\rm
Sq(3)[18]
+ Sq(0,1)[18]}
\\ $h_{5}:$   [1]
\item[19]
{\rm
Sq(1)[21]}
\\ $h_{0}:$   [21]
\\ $h_{1}:$   [19]
\\ $h_{3}:$   [15]
\end{gl}
\end{bdl}
\end{center}
These items describe the locations with 
$(s,t-s) = $ $(10,40)$, $(8,40)$, and $(6,40)$ respectively.

The generators of the resolution are numbered consecutively within each $C_s$,
starting with $h_0^s = [0]$. 
One can see that the 13th generator of $C_{10}$, the 18th generator of $C_8$,
and both the 18th and 19th generators of $C_6$ have topological dimension 40.  

Each generator is first described by its minimal attachment, followed by
the $h_i$ divisibility information. 
For example generator $[18] \in C_8$ is attached to $[12] \in C_7$ by 
${\rm Sq(0,2)}$.

If the minimal attachment is not given completely this is indicated
by ``red.\ mat.''.

The $h_i$ divisibility information in these bidegrees is unrepresentatively
nice; in bidegree $(s,t-s)=(6,40)$ we have 
$[18] = h_5 \cdot [1]$ and 
$[19] = h_0 \cdot [21] = h_1 \cdot [19] = h_3 \cdot [15]$
where numbers on the RHS refer to $C_5$.
In general, if the entry for the generator $[m] \in C_s$ contains
$$
h_i: [n_1], [n_2], \ldots, [n_k]
$$
this means that $[m]$ is a summand of each product $h_i \cdot[n_j]$.
This is illustrated by bidegree $(s,t-s) = (6,62)$:
\begin{bdl}
\item[6/62]
\begin{gl}
\item[34]
{\rm
Sq(7,1)[31]
+ Sq(4,2)[31]
+ Sq(1,3)[31]
+ Sq(3,0,1)[31]
+ Sq(0,1,1)[31]}
\\ $h_{5}:$   [13]
\item[35]
{\rm
Sq(1)[34]
+ Sq(1)[32]}
\\ $h_{0}:$   [32], [34]
\\ $h_{5}:$   [13], [14]
\end{gl}
\end{bdl}
Here
$h_5 \cdot [13] = [34]+[35]$,
$h_5 \cdot [14] = [35]$,
$h_0 \cdot [32] = [35]$, and $h_0 \cdot [34] = [35]$,
where RHS numbers refer to $C_6$, LHS numbers to $C_5$.


\sloppy

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\small
\raggedright

\input{ahssdat}

\end{document}
