A new spectral sequence for homology of posets

Abstract

Abstract. We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding differentials. Our method proves to be powerful and involves far fewer computations than the standard one. We derive many applications of our technique which include a generalization of Hurewicz theorem for regular CW-complexes, results in homological Morse theory and formulas to compute the Möbius function of posets.

Regular coverings and fundamental groupoids of Alexandroff spaces

“…It is well known that Whitehead's Theorem does not hold in the context of Alexandroff spaces, that is, that there exist weak homotopy equivalences between Alexandroff spaces that are not homotopy equivalences. In fact, there exist weakly contractible 6 finite topological spaces which are not contractible. As we show in [5], the minimum cardinality of such a space is 9, and there exist, up to homeomorphism, two weakly contractible non-contractible spaces of 9 points.…”

“…5 M. McCord shows in [14, Lemma 9] that q X is a homotopy equivalence in the usual (topological) sense. 6 A topological space is weakly contractible if all its homotopy groups are trivial.…”

“…Hence, McCord's results show that the problem of computing the homotopy, homology and cohomology groups of Alexandroff spaces is equivalent to the problem of computing the homotopy, homology and cohomology groups of simplicial complexes. In particular, while the computation of the singular homology groups of finite topological spaces can be performed by standard methods (see [6,Section 2]), the example at the end of [14] shows that there exists a space with six points which is weakly equivalent to the sphere S 2 and hence its homotopy groups of high degree remain unknown.…”

Smallest weakly contractible non-contractible topological spaces

“…In [7], we studied the homology groups of finite T 0 -spaces obtaining several results and applications. Among them we mention the following proposition, which will be needed later.…”

Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives

“…See[9] for an interesting approach to the computation of homology groups of finite T 0 -spaces using spectral sequences.…”