Geometric Obstructions in Finsler Spaces and Torsion-Free Persistent Homology

We relate the novel concept of Topological Data Analysis in Finsler space with representability property, which is a natural obstruction to prevent spurious features in high dimensions. We use decomposition of integer matrix in order to find suitable prime integer $p$ such that persistent homology module over ${\mathbb{Z}}_p$ encompasses only the holes associated to the free part, in agreement with the rational case.