Exactness of homotopy functors of spaces

We will provide an analysis of the generalized Atiyah–Hirzebruch spectral sequence (GAHSS), which was introduced by Hakim-Hashemi and Kahn. To do so, we introduce a new class of functors, called nn-exact   functors, which are analogous to Goodwillie’s nn-excisive functors. In the study of these functors, we introduce a new spectral sequence, the homological Barratt–Goerss spectral sequence (HBGSS), which has properties similar to those of the classical Barratt–Goerss Spectral Sequence on homotopy. We close by giving an identification of the E2E2 term of the GAHSS in the case of 2-exact functors on Moore spaces.