Is the 4-line of the E_2 term of the classical Adams spectral sequence known? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:13:22Z http://mathoverflow.net/feeds/question/98229 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98229/is-the-4-line-of-the-e-2-term-of-the-classical-adams-spectral-sequence-known Is the 4-line of the E_2 term of the classical Adams spectral sequence known? cdouglas 2012-05-28T23:14:18Z 2012-05-29T07:33:26Z <p>In other words:</p> <blockquote> <p>What is $\mathrm{Ext}_{\mathcal{A}}^{4,t}(\mathbb{Z}/2,\mathbb{Z}/2)$?</p> </blockquote> <p>If the 4-line is not known, how much is known about it?</p> <p>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.</p> <p>For context, </p> <ul> <li>the 1-line is generated by the classes $h_i$, $i \geq 0$, ($\mathrm{deg}\: h_i = (1,2^i)$), </li> <li>the 2-line is generated by the product classes $h_i h_j$, subject to the relations $h_i h_{i+1} = 0$ and $h_i h_j = h_j h_i$,</li> <li><p>the 3-line is generated by two sets of classes, </p> <ol> <li>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 $h_i h_j = h_j h_i$, </li> <li>the Massey products $\langle h_{i+1},h_i,h_{i+2}^2 \rangle$.</li> </ol></li> </ul> http://mathoverflow.net/questions/98229/is-the-4-line-of-the-e-2-term-of-the-classical-adams-spectral-sequence-known/98254#98254 Answer by Mike-Doherty for Is the 4-line of the E_2 term of the classical Adams spectral sequence known? Mike-Doherty 2012-05-29T07:33:26Z 2012-05-29T07:33:26Z <p>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.</p> <p>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,*}</p>