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Classification of Cellular Fake Surfaces

Lucas Fagan, Yang Qiu, Zhenghan Wang

TL;DR

This work addresses the classification of acyclic cellular fake surfaces, a combinatorial model for 2-dimensional singular polyhedra tied to key topological conjectures. It introduces a rigorous, encoder-driven framework based on $4$-regular 1-skeletons and attaching maps to generate and verify all acyclic examples up to complexity $4$ (with partial data at complexity $5$), and uses determinant tests on boundary maps to certify acyclicity. The authors prove the contractibility conjecture for complexities up to $4$ and establish the embedded-disk conjecture up to complexity $5$ for contractible cases, while providing extensive data and code to support further exploration. Collectively, the results yield explicit catalogs (e.g., 2 surfaces at complexity $1$, 17 at $2$, 238 at $3$, and 4618 at $4$) and reinforce the view that spine prevalence diminishes with complexity, with important implications for Zeeman-type arguments and spine constructions in low-dimensional topology.

Abstract

Generic polyhedra are interesting mathematical objects to study in their own right. In this paper, we initialize a systematic study of two-dimensional generic polyhedra with an eye towards applications to low-dimensional topology, especially the Andrews-Curtis and Zeeman conjectures. After recalling the basic notions of generic polyhedra and fake surfaces, we derive some interesting properties of fake surfaces. Our main result is a complete classification of acyclic cellular fake surfaces up to complexity 4 and a classification of acyclic cellular fake surfaces without small disks of complexity 5. From this classification, we prove the contractibility conjecture for acyclic cellular fake surfaces of complexity 4, and the embedded disk conjecture up to complexity 5. We provide evidence for the conjectures that the probability of being a spine among fake surfaces is 0 and that every contractible fake surface has an embedded disk.

Classification of Cellular Fake Surfaces

TL;DR

This work addresses the classification of acyclic cellular fake surfaces, a combinatorial model for 2-dimensional singular polyhedra tied to key topological conjectures. It introduces a rigorous, encoder-driven framework based on -regular 1-skeletons and attaching maps to generate and verify all acyclic examples up to complexity (with partial data at complexity ), and uses determinant tests on boundary maps to certify acyclicity. The authors prove the contractibility conjecture for complexities up to and establish the embedded-disk conjecture up to complexity for contractible cases, while providing extensive data and code to support further exploration. Collectively, the results yield explicit catalogs (e.g., 2 surfaces at complexity , 17 at , 238 at , and 4618 at ) and reinforce the view that spine prevalence diminishes with complexity, with important implications for Zeeman-type arguments and spine constructions in low-dimensional topology.

Abstract

Generic polyhedra are interesting mathematical objects to study in their own right. In this paper, we initialize a systematic study of two-dimensional generic polyhedra with an eye towards applications to low-dimensional topology, especially the Andrews-Curtis and Zeeman conjectures. After recalling the basic notions of generic polyhedra and fake surfaces, we derive some interesting properties of fake surfaces. Our main result is a complete classification of acyclic cellular fake surfaces up to complexity 4 and a classification of acyclic cellular fake surfaces without small disks of complexity 5. From this classification, we prove the contractibility conjecture for acyclic cellular fake surfaces of complexity 4, and the embedded disk conjecture up to complexity 5. We provide evidence for the conjectures that the probability of being a spine among fake surfaces is 0 and that every contractible fake surface has an embedded disk.
Paper Structure (28 sections, 6 theorems, 26 equations, 10 figures, 3 tables)

This paper contains 28 sections, 6 theorems, 26 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Generic polyhedra and singular manifolds
  3. Generic $n$-polyhedra and $n$-singular manifolds
  4. Fake surfaces
  5. $T$-bundles and spines
  6. Non-spine fake surfaces
  7. Embedded disks
  8. Checking For Embedded Disks and $T$-bundles
  9. Algorithms for deforming 2-complexes into fake surfaces
  10. Mapping class groups of $2$-singular manifolds
  11. Bordism groups of $2$-singular manifolds
  12. Classification of Acyclic Cellular Fake surfaces
  13. Notation and convention
  14. Classification strategy
  15. Classification of 4-regular multigraphs
  16. ...and 13 more sections

Key Result

Theorem 1

There are two acyclic cellular fake surfaces of complexity one, 17 of complexity two, 238 of complexity three, and 4618 of complexity four.

Figures (10)

  • Figure 1: Two Types of Singularities
  • Figure 2: Part of a Neighborhood of an Embedded Disk in a Fake Surface
  • Figure 3: Boundary points
  • Figure 4: Construction of a fake surface
  • Figure 5: Valid Edge Labeling for $\Gamma$
  • ...and 5 more figures

Theorems & Definitions (15)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 7
  • Conjecture 8
  • Definition 9
  • Proposition 10
  • ...and 5 more