A synthetic approach to detecting 𝑣₁-periodic families

We provide a new proof that the unit map from the sphere spectrum to the connective image-of- J J spectrum j j is surjective on homotopy groups. This is done by constructing modified Adams and Adams–Novikov spectral sequences for j j using synthetic spectra. We easily compute these spectral sequences in their entirety using long exact sequences and Adams operations on topological K K -theory. In particular, these computations and deductions are carried out without ever directly calculating F p F_p - or B P BP -homology nor the associated Ext groups of j j nor k o ko .