Algunas aplicaciones de homología simplicial a problemas clásicos de la geometría

Abstract

Algebraic topology, systematically developed by Henri Poincaré at the end of the 19th century, studies topological spaces using algebraic tools. This work focuses on geometric complexes and finite polyhedra through simplicial homology, with the aim of identifying invariant properties such as connectivity, the number of holes, and the underlying topological structure of spaces. The study emphasizes the topological structure of a simplicial complex K, independently of the chosen triangulation or orientation. Using concepts such as the Euler characteristic and Betti numbers, the classification of the five regular polyhedra is explored, showing how these tools allow for the description and comparison of their fundamental properties. Finally, the work highlights that the algebraic machinery of homology provides an effective method for solving classical problems in geometry and topology, offering a more visual and intuitive understanding of the underlying structure of polyhedra and reducing the abstract nature of algebraic topology.