\(v_ 1\)-periodicity in the unstable Adams spectral sequence
In this paper, we show that the unstable Adams spectral sequence (ASS) converging to the 2-primary homotopy groups of the sphere \(S^{2n+1}\) is periodic above a line of slope 1/5, with periodicity operator \(v_ 1\) which moves along lines of slope \(1/2\). We also show that in this region this unstable ASS agrees with the classical ASS for the stable homotopy groups of \(\Sigma^{2n+1}P^{2n}\) with \(P^{2n}\) a real projective space. We give a complete calculation of the \(v_ 1\)-periodic homotopy groups of \(S^{2n+1}\), and we obtain an analogue of Adams' periodicity theorem, which applies to Ext in the category of finite unstable A-modules. Odd-primary analogues of all these results are also obtained.
unstable module over the Steenrod algebra
scientific article; zbMATH DE number 4105868
scientific article