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Foundations of Simplicial Complexes:From Geometric Independence to Realizations

Sanjay Mishra

TL;DR

The paper builds a rigorous, computation-friendly foundation for simplicial complexes, starting from Euclidean spaces and finite point data to define geometric independence, simplices, and simplicial complexes. It develops precise, rank-based criteria for GI, formalizes affine and barycentric structures, and connects these to geometric realizations via a coherent topology. Key contributions include a detailed account of derived structures (subcomplexes, skeleta, stars, links) and topological properties of realizations (Hausdorff, compactness in the finite case, local finiteness). The work emphasizes explicit constructions, examples, and verification methods to bridge combinatorial data with continuous topological spaces, with clear implications for computational topology and related fields.

Abstract

This paper develops a complete foundational treatment of simplicial complexes from Euclidean spaces through geometric realizations, emphasizing concrete computations, examples, and practical verification methods. Beginning with finite point sets in finite and infinite-dimensional Euclidean spaces, geometric independence is established via linear independence of relative vectors, with explicit matrix rank tests. $n$-simplices arise as convex hulls of such independent points, proven convex, compact, uniquely spanned, and homeomorphic to unit balls, with detailed barycentric coordinate. Simplicial complexes form through collections closed under faces and with simplex intersections either empty or common faces, verified by necessary and sufficient disjoint interior conditions, illustrated across dimensions from lines to tetrahedra plus non-examples. Derived structures including subcomplexes, $p$-skeletons, vertices, stars, and links lead to geometric realizations as continuous spaces with weak topology, proven Hausdorff and locally compact, alongside ray characterizations of convexity and continuity via simplicial maps.

Foundations of Simplicial Complexes:From Geometric Independence to Realizations

TL;DR

The paper builds a rigorous, computation-friendly foundation for simplicial complexes, starting from Euclidean spaces and finite point data to define geometric independence, simplices, and simplicial complexes. It develops precise, rank-based criteria for GI, formalizes affine and barycentric structures, and connects these to geometric realizations via a coherent topology. Key contributions include a detailed account of derived structures (subcomplexes, skeleta, stars, links) and topological properties of realizations (Hausdorff, compactness in the finite case, local finiteness). The work emphasizes explicit constructions, examples, and verification methods to bridge combinatorial data with continuous topological spaces, with clear implications for computational topology and related fields.

Abstract

This paper develops a complete foundational treatment of simplicial complexes from Euclidean spaces through geometric realizations, emphasizing concrete computations, examples, and practical verification methods. Beginning with finite point sets in finite and infinite-dimensional Euclidean spaces, geometric independence is established via linear independence of relative vectors, with explicit matrix rank tests. -simplices arise as convex hulls of such independent points, proven convex, compact, uniquely spanned, and homeomorphic to unit balls, with detailed barycentric coordinate. Simplicial complexes form through collections closed under faces and with simplex intersections either empty or common faces, verified by necessary and sufficient disjoint interior conditions, illustrated across dimensions from lines to tetrahedra plus non-examples. Derived structures including subcomplexes, -skeletons, vertices, stars, and links lead to geometric realizations as continuous spaces with weak topology, proven Hausdorff and locally compact, alongside ray characterizations of convexity and continuity via simplicial maps.

Paper Structure

This paper contains 10 sections, 36 theorems, 164 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Euclidean Space
  3. Analytic geometry of Euclidean space
  4. An $n$-Dimensional Plane Spanned by a Geometrically Independent Set
  5. Affine Transformation on ${\mathbb R}^{N}$
  6. Simplices
  7. Simplicial Complex in ${\mathbb R}^N$
  8. Structures derived from Simplicial Complexes
  9. Geometric Realization of Simplicial Complex
  10. Topological Properties of Geometric Realization of Simplicial Complex

Key Result

Theorem 2.8

The Euclidean space ${\mathbb R}^n$ is a vector space over the field ${\mathbb R}$ with vector addition and scalar multiplication for all $\mathbf{x},\mathbf{y} \in {\mathbb R}^n$ and $\alpha \in {\mathbb R}$.

Figures (5)

  • Figure 11: Constructing the 2-simplex $\sigma_2$ from $\bm{a}_0$ and the edge $[\bm{a}_1, \bm{a}_2]$
  • Figure 12: Simplicial Complexes of Increasing Dimension
  • Figure 13: Two examples of 3-dimensional simplicial complexes
  • Figure 14: Non-simplicial Complex
  • Figure 15: Geometric Realization $|\mathcal{K}|$ of Simplicial Complex $\mathcal{K}$ in ${\mathbb R}^2$

Theorems & Definitions (144)

  • Definition 2.1: Finite-Dimensional Euclidean Space as Data Set
  • Definition 2.2: Infinite-Dimensional Euclidean Space
  • Remark 2.3
  • Definition 2.4: Finite Subset of $\mathbb{R}^m$
  • Remark 2.5
  • Definition 2.6: Finite Subset in ${\mathbb R}^N$
  • Definition 2.7: Generalized Euclidean space
  • Example 2.1
  • Example 2.2: Representation of Finite Data Set
  • Theorem 2.8: Euclidean Space as Vector Space
  • ...and 134 more