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 It is worth mentioning that s= 5 is known and the decomposables in s=6 are as well, though this is unpublished.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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.