Iteration of functions and contractibility of acyclic 2-complexes

TL;DR

This paper demonstrates the undecidability of contractibility for certain infinite acyclic 2-complexes and links this property to the Collatz conjecture, highlighting deep connections between topology and number theory.

Contribution

It constructs a specific acyclic 2-complex whose contractibility is equivalent to the truth of the Collatz conjecture, establishing an undecidability result.

Findings

No algorithm can decide contractibility for these complexes.

Constructed complex is contractible iff Collatz conjecture is true.

Links topological properties to a famous number theory conjecture.

Abstract

We show that there can be no algorithm to decide whether infinite recursively described acyclic aspherical 2-complexes are contractible. We construct such a complex that is contractible if and only if the Collatz conjecture holds.