Effective homological computations on finite topological spaces
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Effective homological computations on finite topological spaces Julián Cuevas-Rozo1 · Laureano Lambán2 · Ana Romero2 · Humberto Sarria1 Received: 16 April 2019 / Revised: 19 February 2020 / Accepted: 13 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The study of topological invariants of finite topological spaces is relevant because they can be used as models of a wide class of topological spaces, including regular CWcomplexes. In this work, we present a new module for the Kenzo system that allows the computation of homology groups with generators of finite topological spaces in different situations. Our algorithms combine new constructive versions of well-known results about topological spaces with combinatorial methods used on finite spaces. In the particular case of h-regular spaces, effective and reasonably efficient methods are implemented and the technique of discrete vector fields is applied in order to improve the previous algorithms. Keywords Finite topological spaces · Simplicial complexes · Discrete vector fields · Homology Mathematics Subject Classification 68W30 · 55-04 · 55U15 · 06A07
1 Introduction The study of finite topological spaces is a problem of great interest from a topological point of view, since they can be considered as models for a wide variety of topo-
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Julián Cuevas-Rozo [email protected] Laureano Lambán [email protected] Ana Romero [email protected] Humberto Sarria [email protected]
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Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
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Departamento de Matemáticas y Computación, Universidad de La Rioja, Logroño, Spain
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J. Cuevas-Rozo et al.
logical spaces, including regular CW-complexes. Also, these spaces have been used to describe objects in image processing [11], [12] and for evaluating the topological inconsistency of geospatial data in [10]. Therefore, it is important to provide tools, both theoretical and computational, allowing to determine topological invariants of finite topological spaces. There exists an interesting variety of tools which have been developed for the study of finite topological spaces connecting them with different mathematical structures such as pre-ordered sets [1], simplicial complexes [13], topogenous matrices [21] and, in recent years, they have been related with submodular functions [5], enriching the study of topological structures with different points of view. In this way, the use of finite topological spaces allows the combination of techniques coming from the theory of topology with other methods of combinatorial nature; this is the line that we pretend to consider in this work. In general, symbolic computation systems in topology are devoted to computing invariants of topological spaces. Algebraic topology provides a linearization of topological problems, which usually present a pure non-linear nature. Then, the pursuit of reasonable linear algebra algorithmic solutions is of great interest from a computational point of view. In particular, our interest co
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