We show that if we factor the long exact sequence in cohomology of a cofiber sequence of spectra into short exact sequences, then the d_2-differential in the Adams spectral sequence of any one term is related in a precise way to Yoneda composition with the 2-extension given by the complementary terms in the long exact sequence. We use this to give a complete analysis of the Adams spectral sequence for the connective image-of-J spectrum, finishing a calculation that was begun by D. Davis [Bol. Soc. Mat. Mexicana (2) 20 (1975), pp. 6–11].
Algebraic Geometry | Geometry and Topology
Bruner, R. & Rognes, J. The Adams spectral sequence for the image-of-J spectrum. Trans. Amer. Math. Soc. 375 (2022), 5803-5827. https://doi.org/10.1090/tran/8680