On homology of finite topological spaces

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.

28 pages. A new section was added (section 5). And some minor changes were performed

References in corpus (3)

• $G$-colorings of posets, coverings and presentations of the fundamental group
• Gradings, smash products and Galois coverings of a small category
• The second homotopy group in terms of colorings of locally finite models and new results on asphericity

Cited by in corpus (6)

• Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives
• Regular coverings and fundamental groupoids of Alexandroff spaces
• Smallest homotopically trivial non-contractible spaces
• Homology of posets with functor coefficients and its relation to Khovanov homology of knots
• (Co)homology theories for structured spaces arising from their corresponding poset
• Homology of Finite Topological Spaces