A new spectral sequence for homology of posets (Q501585)

In [\textit{P. Alexandroff}, Rec. Math. Moscou, n. Ser. 2, 501--519 (1937; Zbl 0018.09105)] a correspondence is described between finite posets and finite \(T_0\)-spaces. Insights into applications of finite topological spaces have resulted in their study developing considerably in recent years. The authors of the current work expound: ``As it is shown by the recent works of Barmak and Minian, finite topological spaces can be used in several different situations to develop new tools and techniques to study topological and combinatorial problems.'' Conversely, standard techniques from homotopy theory can be used to study finite topological spaces, as we see in the article under review. The main result is Theorem 3.10, wherein a new spectral sequence converging to the homology groups of a given finite space is produced. This theorem, in conjunction with a remark following its proof, completely and explicitly describes the differentials on every page of said spectral sequence. The authors emphasize the simplicity (in relation to existing approaches) and explicitness of their result, with detailed examples being discussed throughout the development. A version of Theorem 3.10 for relative homology groups is given in Theorem 3.17. The final section highlights Theorem 4.3, in which the authors use their new techniques to generalize Theorem 3.14 of \textit{E. G. Minian} [Topology Appl. 159, No. 12, 2860--2869 (2012; Zbl 1258.57012)] regarding discrete Morse theory, providing a ``completely different and more conceptual proof.''