Is the 4-line of the E_2 term of the classical Adams spectral sequence known?

2012-05-28T23:14:18Z

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 $\langle h_{i+1},h_i,h_{i+2}^2 \rangle$.

Answer by Mike-Doherty for Is the 4-line of the E_2 term of the classical Adams spectral sequence known?

2012-05-29T07:33:26Z

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,*}

Is the 4-line of the E_2 term of the classical Adams spectral sequence known? - MathOverflow

Is the 4-line of the E_2 term of the classical Adams spectral sequence known?

Answer by Mike-Doherty for Is the 4-line of the E_2 term of the classical Adams spectral sequence known?