The Hurwitz Equivalence Problem is Undecidable

TL;DR

This paper proves that determining Hurwitz equivalence for certain factorizations in algebraic structures and braid groups is fundamentally undecidable, highlighting limits of algorithmic approaches in these areas.

Contribution

It establishes the undecidability of the Hurwitz equivalence problem for specific algebraic and braid group factorizations, extending known undecidability results.

Findings

Hurwitz equivalence problem is undecidable for 1-factorizations in F_2 ⊕ F_2

Undecidability extends to Δ^2-factorizations in braid groups B_n for n ≥ 5

Results demonstrate fundamental limits of algorithmic solutions in algebraic topology and group theory

Abstract

In this paper, we prove that the Hurwitz equivalence problem for 1-factorizations in $F_2 \oplus F_2$ is undecidable, and as a consequence, the Hurwitz equivalence problem for $\Delta^2$-factorizations in the braid groups $B_n, n\geq 5$ is also undecidable.