Matrix Invariants as Homotopy Invariants in Finite $T_0$-spaces

TL;DR

This paper introduces a novel approach linking finite $T_0$-spaces to matrix invariants, providing simple homotopy invariants for classifying finite topological spaces and simplicial complexes.

Contribution

It establishes a bijection between finite $T_0$-spaces and matrix classes, using determinants and ranks as homotopy invariants, advancing the algebraic understanding of finite topological spaces.

Findings

Determinant and rank serve as homotopy invariants.

Bijection between finite $T_0$-spaces and matrix classes.

New insights into finite posets and their matrix representations.

Abstract

We establish a bijection between the set of finite topological $T_0$-spaces (or partially ordered sets) and equivalence classes of square matrices. The absolute value of the determinant or the rank of these matrices serve as simple homotopy invariants for the corresponding topological spaces, and consequently, for finite simplicial complexes. To conclude, we explore further relationships and problems concerning finite posets within the context of these matrices.