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On piecewise linear cell decompositions

Alexander Kirillov

TL;DR

The paper introduces PLCW complexes, a structured generalization of triangulations that lies between triangulations and CW complexes, tailored for PL topology and extended TQFT frameworks. It formalizes generalized cells and PLCW attachments via regular cellular maps, and proves that radial and elementary subdivisions preserve the PLCW structure, enabling controlled refinements. The central result is an Alexander-type theorem: any two PLCW decompositions of the same polyhedron are related by a finite sequence of elementary subdivisions (the (n,n) moves suffice). This provides a robust combinatorial toolkit for state-sum invariants and extended topological quantum field theories, linking concrete cell decompositions with higher-categorical and quantum-field-theoretic constructions.

Abstract

In this note, we introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be obtained from each other by a sequence of certain "elementary" moves. This definition is motivated by the needs of Topological Quantum Field Theory, especially extended theories as defined by Lurie.

On piecewise linear cell decompositions

TL;DR

The paper introduces PLCW complexes, a structured generalization of triangulations that lies between triangulations and CW complexes, tailored for PL topology and extended TQFT frameworks. It formalizes generalized cells and PLCW attachments via regular cellular maps, and proves that radial and elementary subdivisions preserve the PLCW structure, enabling controlled refinements. The central result is an Alexander-type theorem: any two PLCW decompositions of the same polyhedron are related by a finite sequence of elementary subdivisions (the (n,n) moves suffice). This provides a robust combinatorial toolkit for state-sum invariants and extended topological quantum field theories, linking concrete cell decompositions with higher-categorical and quantum-field-theoretic constructions.

Abstract

In this note, we introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be obtained from each other by a sequence of certain "elementary" moves. This definition is motivated by the needs of Topological Quantum Field Theory, especially extended theories as defined by Lurie.

Paper Structure

This paper contains 9 sections, 10 theorems, 15 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Generalized cells
  4. Generalized cell complexes
  5. PLCW complexes
  6. Subdivisions
  7. Elementary subdivisions
  8. Main theorem
  9. Orientations

Key Result

Lemma 3.2

If $C$ is a convex cell and $f\colon C\to \R^m$ is regular, then $C$ admits a cell decomposition $K$ such that for any cell $K_i\in K$, the restriction $f|_{K_i}$ is injective.

Figures (5)

  • Figure 3: An example of a regular cellular map. It identifies edges $a_1b_1$ and $a_2b_2$, sending each of them to edge $ab$.
  • Figure 4: Examples of radial subdivisions. Note that in the last example, we are subdividing a 1-cell.
  • Figure 5: An elementary subdivision
  • Figure 6: A non-elementary subdivision
  • Figure 7: Obtaining a non-elementary subdivision by a sequence of elementary subdivisions and their inverses

Theorems & Definitions (29)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Theorem 3.4
  • proof
  • Definition 4.1
  • Example 4.2
  • Definition 4.3
  • Definition 5.1
  • ...and 19 more