A new family of filtration three in the stable homotopy of spheres
Let \(p\) be a prime greater than 3. In the Adams-Novikov spectral sequence \[ E(BP)^*_2= \text{Ext}^*_{BP_*(BP)} (BP_*,BP_*) \Rightarrow \pi_*(S), \] we have a relation \(\alpha_1 \gamma_{p^{n-2}/p^{n-2}-1} =\beta_1\beta_{p^{n-1} /p^{n-1}-1}\in E(BP)^4_2\), which makes us to ask if \(\gamma_{p^{n-2}/p^{n-2}-1}\) is a permanent cycle. The main result of this paper is that each element \(b_0h_n +h_1b_{n-1}\in E^3_2\) for \(n\geq 3\) is a permanent cycle in the Adams spectral sequence \(E^*_2= \text{Ext}^*_A (\mathbb{Z}/p,\mathbb{Z}/p)\) \(\Rightarrow\pi_*(S).\) This gives an affirmative answer of the above question if we replace \(\gamma_{ p^{n-2}/p^{n-2}-1}\) by \(\gamma_{p^{n-2}/p^{n-2}-1} +\cdots\in E_2 (BP)^3\). Indeed, the Thom reduction \(\Phi:E (BP)^3_2\to E^3_2\) acts as \(\Phi (\gamma_{ p^{n-2} /p^{n-2}-1})= b_0h_n+h_1 b_{n-1}\). The proof is based on the existence of Cohen's elements \(\zeta_{n-1} \in\pi_*(S)\) \((n\geq 2)\) detected by \(h_0b_{n-1} \in E^3_2\). Using the elements, the author showed first that \(h_0 h_n\) is a permanent cycle in the Adams spectral sequence \(E^*_r(M)\) for \(\pi_*(M)\), where \(M\) is the mod \(p\) Moore spectrum, and then he showed that \(h_1 h_n\) is permanent in \(E^*_r(M)\) by observing the Adams resolution, which is a hard part. The main result follows as the image \(\delta (h_1h_n) =b_0h_n+h_1 b_{n-1}\) under the connecting homomorphism \(\delta: E^2_2(M) \to E^3_2\).
scientific article; zbMATH DE number 1741730