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

Pedro J. Chocano

TL;DR

The paper develops a matrix-centric framework that encodes finite $T_0$-spaces (posets) as 0-1 square matrices, establishing a bijection with matrix equivalence classes and showing that simple homotopy invariants such as $| ext{det}(X_M)|$ and $ar{ ext{rank}}(X)$ arise naturally from these matrices. It connects these invariants to the reduced Euler characteristic via the functors between finite spaces and simplicial complexes, and analyzes when other matrix invariants (e.g., the characteristic polynomial) reflect topology or homotopy. By interpreting matrices as adjacency-like objects of associated digraphs and studying group actions and summation invariants, the work reveals both powerful tools and limitations for distinguishing finite spaces and their homotopy types. Overall, the results provide a concrete linear-algebraic approach to finite-space topology, linking posets, graphs, and simple homotopy theory through matrix invariants.

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.

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

TL;DR

The paper develops a matrix-centric framework that encodes finite -spaces (posets) as 0-1 square matrices, establishing a bijection with matrix equivalence classes and showing that simple homotopy invariants such as and arise naturally from these matrices. It connects these invariants to the reduced Euler characteristic via the functors between finite spaces and simplicial complexes, and analyzes when other matrix invariants (e.g., the characteristic polynomial) reflect topology or homotopy. By interpreting matrices as adjacency-like objects of associated digraphs and studying group actions and summation invariants, the work reveals both powerful tools and limitations for distinguishing finite spaces and their homotopy types. Overall, the results provide a concrete linear-algebraic approach to finite-space topology, linking posets, graphs, and simple homotopy theory through matrix invariants.

Abstract

We establish a bijection between the set of finite topological -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.

Paper Structure

This paper contains 11 sections, 39 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite Spaces, Matrices and Graphs
  4. Invariants
  5. Determinant
  6. Rank
  7. Characteristic polynomial and eigenvalues
  8. Other aspects of the matrices from $\mathcal{M}$
  9. Determinant of the sum of a topological space with the identity
  10. Group actions
  11. Sums of columns and rows as topological invariants

Key Result

Theorem 2.3

Let $X$ be a finite $T_0$-space and let $x\in X$ be a beat point. Then $X\setminus \{x\}$ is a strong deformation retract of $X$.

Theorems & Definitions (70)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: stong1966finite
  • Definition 2.4
  • Theorem 2.5: stong1966finite
  • Definition 2.6
  • Theorem 2.7: barmak2008simple
  • Definition 2.8
  • Theorem 2.9: mccord1966singular
  • Definition 2.10
  • ...and 60 more