Effectiveness of Walker's Cancellation Theorem

TL;DR

This paper investigates the computational complexity of effectively constructing isomorphisms in Walker's Cancellation theorem for finitely generated abelian groups, establishing that the process's complexity is equivalent to the halting problem.

Contribution

It extends Deveau's analysis by demonstrating that the uniform output of isomorphism indices has a complexity of ' (the halting problem), clarifying the computational limits.

Findings

The complexity of uniformly outputting an isomorphism is '

The process cannot be made uniform without reaching the halting problem's complexity

Provides a detailed analysis of the computational aspects of Walker's Cancellation theorem.

Abstract

Walker's Cancellation theorem for abelian groups tells us that if $A$ is finitely generated and $G$ and $H$ are such that $A \oplus G \cong A \oplus H$, then $G \cong H$. Michael Deveau showed that the theorem can be effectivized, but not uniformly. In this paper, we expand on Deveau's initial analysis to show that the complexity of uniformly outputting an index of an isomorphism between $G$ and $H$, given indices for $A$, $G$, $H$, the isomorphism between $A \oplus G$ and $A \oplus H$, and the rank of $A$, is $\mathbf{0'}$.