Effective computation of invariants of finite topological spaces

Résumé

In this work we have presented effective algorithms to compute invariants of finite topological spaces. These algorithms have been developed by combining combinatorial techniques on posets, which have been stated in the foundational papers of the theory of finite spaces, and recent results in this line of work, such us methods to maintain weak homotopy types or simple homotopy types and the application of Discrete Morse Theory. The memoir has been organized in five chapters. Chapter 1 contains some notations, definitions and previous results about Algebraic Topology on finite topological spaces, their connections with simplicial complexes and some insights into the Discrete Morse Theory, which will be useful along our work. Also, a presentation of the Kenzo program as a powerful tool for the development of our algorithmic implementations in posterior chapters, has been included. Up to now, the known methods for computing invariants of finite topological spaces were applicable only for face posets of simplicial complexes or regular CW-complexes. In Chapter 2, we have made constructive some theoretical results on finite topological spaces by different authors, producing in particular new algorithms for computing in an explicit way some chain complexes associated with h-regular finite topological spaces (smaller than the chain complex of the order complex of the finite space) and their corresponding generators. We have implemented the previous algorithms in the computer algebra system Kenzo. Up to our knowledge, our new program is the only software able to compute homology groups of finite topological spaces working directly on the posets without having to go, necessarily, to the simplicial world. Moreover, we improve our algorithms on h-regular spaces by using discrete vector fields. In our case, we produce a new algorithm constructing a discrete vector field defined directly on the poset that can be applied to general h-regular finite spaces; as before, up to our knowledge there does not exist any other software producing this kind of construction over general finite topological spaces. The algorithms to compute homology above mentioned are applicable to h-regular finite spaces. In the literature there are few examples of hregular finite spaces, different from face posets of simplicial complexes. The h-regularization process we have described in Chapter 3, produces a wide variety of h-regular finite spaces. Indeed, as we have shown, any finite space of height at most two can be h-regularized, allowing to consider new examples of this kind of spaces. In Chapter 4, we have presented an interface between the computer algebra systems SageMath and Kenzo. Our work has made it possible to work with Kenzo in a friendlier way and to allow both systems to collaborate in some computations which can not be done independently in any of the programs. Moreover, we have created a module implementing finite topological spaces and related concepts in SageMath by using our previously explained Kenzo algorithms. Finally, in Chapter 5, we have considered some strategies trying to study alternatives to compute longer discrete vector fields on finite spaces. Moreover, we consider some machine learning techniques such as reinforcement learning and Monte- Carlo tree search to obtain discrete vector fields as big as possible.