On the unstable homotopy spectral sequences (Q1085483)

The authors study the spectral sequence that arises from the homotopy classes of maps \([S^ kY,X]\), either by a filtration coming from the skeleta of the domain, or that coming from the Postnikov terms of the range. Such spectral sequences have been studied by several authors. \textit{C. R. F. Maunder} [Proc. Camb. Philos. Soc. 59, 567-574 (1963; Zbl 0116.146)] treated the case where the range is an \(\Omega\)-spectrum. The reviewer [Am. J. Math. 94, 1131-1154 (1972; Zbl 0254.55021)] studied the case of a finite complex, but with the additional module structure over the stable homotopy ring of spheres. Working with connected CW-complexes and simple target spaces, the present authors prove the isomorphism of the spectral sequences, coming from the different filtrations, for \(r>2\). They then give applications, beginning with a theorem about the image of differentials lying in torsion subgroups (related to the finite-order of certain k-invariants), and ending with an application of the famous result of \textit{H. Miller} [Ann. Math., II. Ser. 120, 39-87 (1984; Zbl 0552.55014)].