Wen-Hsiung Lin

## The stable homotopy elements $$\{g_3\}\in \pi^S_{92}$$

slide: PDF.
video: comming soon.
abstract:
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.