A Topological proof that $O_2$ is $2$-MCFL

TL;DR

This paper presents a new, dimension-agnostic proof that the group language $O_2$ is 2-multiple context free, potentially extending to higher dimensions, unlike previous proofs specific to two dimensions.

Contribution

It provides a topological proof of Salvati's theorem that does not rely on two-dimensional-specific ideas, opening avenues for generalization.

Findings

New proof of $O_2$ being 2-MCFL without dimension-specific arguments

Potential for generalizing the proof to higher dimensions $O_n$

Simplifies understanding of the topological properties of $O_2$

Abstract

We give a new proof of Salvati's theorem that the group language $O_2$ is $2$ multiple context free. Unlike Salvati's proof, our arguments do not use any idea specific to two-dimensions. This raises the possibility that the argument might generalize to $O_n$.