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Singular homology of roots of unity

Nikola Milićević

TL;DR

This paper develops singular homology for Čech closure spaces, extending classical homology concepts from topological spaces to non-topological, discrete-like settings. The authors establish fundamental results—excision, Mayer–Vietoris, and a dimension-1 Hurewicz theorem—in the closure-space context and apply them to compute homology for spaces such as roots of unity endowed with distance-based closure operators. The approach relies on cone constructions, linear and barycentric simplices, and a theory of small singular chains to prove MV and excision; these tools enable calculations on graph-like structures and pave the way for building larger closure-space homology from simpler components. The work also analyzes how homology behaves under suspensions and under changes to the unity structure, providing partial classifications and isomorphisms useful for handling finite closure spaces and their quotients. Overall, the results offer a robust framework for algebraic topology beyond traditional spaces, with concrete computations for roots of unity and implications for graphs and other combinatorial objects in applied topology.

Abstract

We extend some basic results from the singular homology theory of topological spaces to the setting of Čech's closure spaces. We prove analogues of the excision and Mayer-Vietoris theorems and the Hurewicz theorem in dimension one. We use these results to calculate examples of singular homology groups of spaces that are not topological but are often encountered in applied topology, such as simple undirected graphs. We focus on the singular homology of roots of unity with closure structures arising from considering nearest neighbors. These examples can then serve as building blocks along with our Mayer-Vietoris and excision theorems for computing the singular homology of more complex closure spaces.

Singular homology of roots of unity

TL;DR

This paper develops singular homology for Čech closure spaces, extending classical homology concepts from topological spaces to non-topological, discrete-like settings. The authors establish fundamental results—excision, Mayer–Vietoris, and a dimension-1 Hurewicz theorem—in the closure-space context and apply them to compute homology for spaces such as roots of unity endowed with distance-based closure operators. The approach relies on cone constructions, linear and barycentric simplices, and a theory of small singular chains to prove MV and excision; these tools enable calculations on graph-like structures and pave the way for building larger closure-space homology from simpler components. The work also analyzes how homology behaves under suspensions and under changes to the unity structure, providing partial classifications and isomorphisms useful for handling finite closure spaces and their quotients. Overall, the results offer a robust framework for algebraic topology beyond traditional spaces, with concrete computations for roots of unity and implications for graphs and other combinatorial objects in applied topology.

Abstract

We extend some basic results from the singular homology theory of topological spaces to the setting of Čech's closure spaces. We prove analogues of the excision and Mayer-Vietoris theorems and the Hurewicz theorem in dimension one. We use these results to calculate examples of singular homology groups of spaces that are not topological but are often encountered in applied topology, such as simple undirected graphs. We focus on the singular homology of roots of unity with closure structures arising from considering nearest neighbors. These examples can then serve as building blocks along with our Mayer-Vietoris and excision theorems for computing the singular homology of more complex closure spaces.
Paper Structure (28 sections, 58 theorems, 76 equations, 3 figures)

This paper contains 28 sections, 58 theorems, 76 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Closures and interiors
  4. Continuous functions
  5. Covers of closure spaces
  6. Limits and colimits of closure spaces
  7. CW complexes in $\mathbf{Cl}$
  8. Homotopy for closure spaces
  9. Homotopy
  10. Homotopy extension property.
  11. The fundamental group
  12. Singular homology for closure spaces
  13. Hurewicz theorem in dimension $1$ for closure spaces
  14. Excision and Mayer-Vietoris theorems for closure spaces
  15. Cone construction
  16. ...and 13 more sections

Key Result

Theorem 1

If $X$ is path-connected $h$ is surjective and its kernel is the commutator subgroup of $\pi_1(X)$. Hence $H_1(X)$ is the abelianization of $\pi_1(X)$.

Figures (3)

  • Figure 1: Homotopy $H$ as a sum of two singular $2$-simplices.
  • Figure 2: The singular $1$-chain $f_2-f_1\cdot f_2+f_1$ as a boundary of a singular $2$-simplex.
  • Figure 3: The closure space $(\mathbb{Z}_{11},c_3)$ (left) and its quotient that is homeomorphic to $(\mathbb{Z}_{7},c_2)$ (right).

Theorems & Definitions (101)

  • Theorem : \ref{['theorem:homology_is_abelianization_of_fundamental_group']}
  • Proposition : \ref{['prop:homology_of_covering_system']}
  • Theorem : \ref{['theorem:excision']}
  • Theorem : \ref{['theorem:Mayer_Vietoris']}
  • Theorem : \ref{['theorem:homology_group_of_circle']}
  • Theorem : \ref{['theorem:homology_of_roots_of_unity']}
  • Theorem : \ref{['theorem:homology_wedge_of_circles']}
  • Theorem : \ref{['theorem:homology_wedge_of_roots_of_unity']}
  • Theorem : \ref{['theorem:higher_homology_for_n_greater_than_4m']}
  • Theorem : \ref{['theorem:higher_homology_for_n_between_3m_and_4m_part_1']}
  • ...and 91 more