Wen-Hsiung Lin

The stable homotopy elements \(\{g_3\}\in \pi^S_{92}\)

slide: PDF.
video: comming soon.
Consider the mod \(2\) Adams spectral sequence \(\{E^{*,*}_r\}\) for the \(2\)-primary stable homotopy groups of spheres \(\pi^S_*\). Its \(E_2\)-term is \(\mathrm{Ext}^{s,t}_{A}(\mathbb{Z}/2,\mathbb{Z}/2)\), \(s\ge0\), \(t\ge0\), where \(A\) denotes the mod \(2\) Steenrod algebra. We simply write \(\mathrm{Ext}^{s,t}_A\) for \(\mathrm{Ext}^{s,t}_{A}(\mathbb{Z}/2,\mathbb{Z}/2)\)
For each \(i\ge0\), \(\mathrm{Ext}^{4,2^{i+4}+2^{i+3}}_{A}\cong \mathbb{Z}/2\) generated by an element called \(g_{i+1}\) ([1]). The first two \(g_1\in \mathrm{Ext}^{4,24}_{A}\) and \(g_2\in \mathrm{Ext}^{4,48}_{A}\) in the family \(\{g_{i+1} \mid i\ge 0\}\) are known to detect homotopy elements: Mimura and Toda's element \(\bar\kappa \in \pi^S_{20}\) is detected by \(g_1\) and Barratt, Mahowald and Tangora have shown that there are elements in \(\pi^S_{44}\) detected by \(\{g_2\}\).
In this talk I will show that \(g_3\in \mathrm{Ext}^{4,96}_{A}\) also detects homotopy elements in \(\pi^S_{92}\).
[1] Wen-Hsiung Lin, \(\mathrm{Ext}^{4,*}_A(\mathbb{Z}/2,\mathbb{Z}/2)\) and \(\mathrm{Ext}^{5,t}_A(\mathbb{Z}/2,\mathbb{Z}/2)\), Topology Appl. (155) (2008), no.5, 459-496.