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Simplicial complexes, stellar moves, and projective amalgamation

Sławomir Solecki

TL;DR

The paper establishes a novel amalgamation class built from stellar subdivision and welding, reframing stellar moves as weld-division maps and proving a projective amalgamation property. It then places this class within projective Fraïssé theory to obtain a canonical Fraïssé limit whose quotient space recovers the geometric realization of the starting complex, providing a combinatorial construction of a topological object with topological dimension exceeding one. Central to the work is a purely combinatorial, finite-sequence calculus on Fin$^+$-structured faces, enabling a non-geometric proof of amalgamation and a robust description of the canonical quotient. The results yield a new bridge between combinatorial topology and model-theoretic limits, with potential implications for understanding geometric realizations through purely combinatorial data.

Abstract

We explore connections between stellar moves on simplicial complexes (these are fundamental operations of combinatorial topology) and projective Fra{ï}ss{é} limits (this is a model theoretic construction with topological applications). We identify a class of simplicial maps that arise from the stellar moves of welding and subdividing. We call these maps weld-division maps. The core of the paper is the proof that the category of weld-division maps fulfills the projective amalgamation property. This gives an example of an amalgamation class that substantially differs from known classes. The weld-division amalgamation class naturally gives rise to a projective Fra{ï}ss{é} class. We compute the canonical limit of this projective Fra{ï}ss{é} class and its canonical quotient space. This computation gives a combinatorial description of the geometric realization of a simplicial complex and an example of a combinatorially defined projective Fra{ï}ss{é} class whose canonical quotient space has topological dimension strictly bigger than $1$. The method of proof of the amalgamation theorem is new. It is not geometric or topological, but rather it consists of combinatorial calculations performed on finite sequences of finite sets and functions among such sequences. Set theoretic nature of the entries of the sequences is crucial to the arguments.

Simplicial complexes, stellar moves, and projective amalgamation

TL;DR

The paper establishes a novel amalgamation class built from stellar subdivision and welding, reframing stellar moves as weld-division maps and proving a projective amalgamation property. It then places this class within projective Fraïssé theory to obtain a canonical Fraïssé limit whose quotient space recovers the geometric realization of the starting complex, providing a combinatorial construction of a topological object with topological dimension exceeding one. Central to the work is a purely combinatorial, finite-sequence calculus on Fin-structured faces, enabling a non-geometric proof of amalgamation and a robust description of the canonical quotient. The results yield a new bridge between combinatorial topology and model-theoretic limits, with potential implications for understanding geometric realizations through purely combinatorial data.

Abstract

We explore connections between stellar moves on simplicial complexes (these are fundamental operations of combinatorial topology) and projective Fra{ï}ss{é} limits (this is a model theoretic construction with topological applications). We identify a class of simplicial maps that arise from the stellar moves of welding and subdividing. We call these maps weld-division maps. The core of the paper is the proof that the category of weld-division maps fulfills the projective amalgamation property. This gives an example of an amalgamation class that substantially differs from known classes. The weld-division amalgamation class naturally gives rise to a projective Fra{ï}ss{é} class. We compute the canonical limit of this projective Fra{ï}ss{é} class and its canonical quotient space. This computation gives a combinatorial description of the geometric realization of a simplicial complex and an example of a combinatorially defined projective Fra{ï}ss{é} class whose canonical quotient space has topological dimension strictly bigger than . The method of proof of the amalgamation theorem is new. It is not geometric or topological, but rather it consists of combinatorial calculations performed on finite sequences of finite sets and functions among such sequences. Set theoretic nature of the entries of the sequences is crucial to the arguments.

Paper Structure

This paper contains 59 sections, 67 theorems, 533 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background and statements of the main results
  3. Subdivided simplicial complexes and maps among them
  4. Stellar subdivision
  5. A class of complexes and maps among them
  6. Additive families of faces
  7. Stellar subdivisions of simplicial maps
  8. Weld-division maps
  9. Statements of the main theorems
  10. Set theoretic setup for complexes and their divisions
  11. The family of sets ${\rm Fin}^+$
  12. Grounded and divided complexes and grounded simplicial maps
  13. Complexes in ${\rm Fin}^+$ and their divisions
  14. Divided complexes and grounded simplicial maps among them
  15. Internal description of faces of divided complexes
  16. ...and 44 more sections

Key Result

Lemma 1.1

Let $S$ be an additive family in a simplicial complex $A$. Let $\vec{S}$ and $\vec{S}'$ be two non-decreasing enumerations of $S$. Then $\vec{S} A=\vec{S}' A$.

Figures (6)

  • Figure 2.1: A weld map with $p=b$, $t = \{a,b,c\}$, and $A=$ all non-empty subsets of $\{a, b, c\}$
  • Figure 2.2: A division of a weld map with $p=b$, $s = \{a,b\}$, $t= \{ a, b, c\}$, and $A=$ all non-empty subsets of $\{a, b, c\}$
  • Figure 9.1: A type 1 isomorphism with $r = \{a\}$, $s = \{b\}$, $t = \{a,b,c\}$
  • Figure 9.2: A type 2 isomorphism with $s = \{a,c\}$, $t = \{b,c\}$
  • Figure 9.3: Type 3a and 3b isomorphisms
  • ...and 1 more figures

Theorems & Definitions (135)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 125 more