In other words:

What is $\mathrm{Ext}_{\mathcal{A}}^{4,t}(\mathbb{Z}/2,\mathbb{Z}/2)$?

If the 4-line is not known, how much is known about it?

Here, $\mathcal{A}$ is the 2-primary Steenrod algebra, $4$ is the homological degree corresponding to the Adams filtration, and $t$ is the internal grading degree. Those $\mathrm{Ext}$ groups make up the fourth row of the classical Adams spectral sequence $E_2 = \mathrm{Ext}_{\mathcal{A}}^{s,t}(\mathbb{Z}/2,\mathbb{Z}/2)$ converging to the 2-adic completion of the $(t-s)^{\mathrm{th}}$ stable homotopy group of the sphere.

For context,

  • the 1-line is generated by the classes $h_i$, $i \geq 0$, ($\mathrm{deg}\: h_i = (1,2^i)$),
  • the 2-line is generated by the product classes $h_i h_j$, subject to the relations $h_i h_{i+1} = 0$ and $h_i h_j = h_j h_i$,
  • the 3-line is generated by two sets of classes,

    1. the product classes $h_i h_j h_k$, subject to the relations implied by $h_i h_{i+2}^2 = 0$, $h_{i+1}^3 = h_i^2 h_{i+2}$, $h_i h_{i+1} = 0$, and $h_i h_j = h_j h_i$,
    2. the Massey products $\langle h_{i+1},h_i,h_{i+2}^2 \rangle$.

2 Answers 2


The 4-line is determined by Wen-Hsiung Lin in "$Ext_A^{4,*}({\bf Z}/2,{\bf Z}/2) $ and $Ext_A^{5,*}({\bf Z}/2,{\bf Z}/2) $", Topology and its Applications (2008) vol 155 no.5 pp 459-496.

He gives a basis for the indecomposable elements in $Ext_A^{4,*}$ and generators and relations for the quotient of $Ext_A^{s,*} $ for $s \le 4$ by the indecomposables of $Ext_A^{4,*}

  • 1
    $\begingroup$ A rare happy moment when there is a preexisting paper devoted precisely to answering the question. Thanks! $\endgroup$
    – cdouglas
    May 31, 2012 at 10:38

This old question caught my eye while looking at another question. It is worth mentioning that s= 5 is known and the decomposables in s=6 are as well, though this is unpublished.

Tai-Wei Chen, Determination of $Ext^{5,∗}(Z/2,Z/2)$, Topology Appl. 158 (2011), no. 5, 660–A689, DOI 10.1016/j.topol.2011.01.002.

Tai-Wei Chen, The structure of decomposable elements in $Ext^{6,∗}(Z/2,Z/2)$, (2012) preprint.


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