Differentials of certain homology Atiyah–Hirzebruch spectral sequences

In this paper, we establish explicit formulas for the differentials in the homology Atiyah–Hirzebruch spectral sequences computing homotopy groups and [Formula: see text] ([Formula: see text])-homology groups. While the differentials in the [Formula: see text]-cohomology Atiyah–Hirzebruch spectral sequence are well-documented, their direct adaptation to the [Formula: see text]-homology spectral sequence remains non-obvious. In this work, we prove that the differentials in the [Formula: see text]-homology spectral sequence admit analogous expressions to their [Formula: see text]-cohomology counterparts, thereby completing this correspondence.