On the odd-primary Adams 2-line elements (Q1962077)

This paper is concerned with the existence of elements in the odd primary stable homotopy groups of spheres which correspond to elements on the \(2\)-line of the Adams spectral sequence. It is known that up to finite exceptions the only possible elements are \(\theta_j\) and \(\eta_j\) corresponding to the \(E_2\) term elements \(b_j=\langle h_j,\dots,h_j\rangle\) and \(h_0h_j\) respectively. The author proves that \(\theta_j\) exists and factors through the double transfer only if \(j \leq 2\) for \(p =3\) or \(j=0\) for \(p \geq 5\) and that if the element \(\eta_j\) exists then it has a nonzero multiple of \(\beta_1\) as its James-Hopf invariant. Some consequences of these results linking them to the Hopkins-Miller spectrum \(EO\) are also presented.