Foundations of Simplicial Complexes:From Geometric Independence to Realizations

TL;DR

This paper provides a comprehensive foundational framework for simplicial complexes in Euclidean spaces, emphasizing geometric independence, realizations, and practical verification methods with detailed examples.

Contribution

It introduces explicit matrix rank tests for geometric independence and detailed methods for constructing and verifying simplicial complexes and their realizations.

Findings

Established linear independence via matrix rank tests.

Proved convexity, compactness, and homeomorphism of simplices.

Verified conditions for simplicial complex intersections and realizations.

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.