Classification of Cellular Fake Surfaces

TL;DR

This paper systematically classifies acyclic cellular fake surfaces up to certain complexities, providing evidence for conjectures related to their topological properties and applications in low-dimensional topology.

Contribution

It offers a complete classification of acyclic cellular fake surfaces up to complexity 4 and partial classification up to complexity 5, advancing understanding of their topological structure.

Findings

Contractibility conjecture proven for complexity 4

Embedded disk conjecture verified up to complexity 5

Probability of fake surfaces being spines is zero

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.