Undecidability problems for semifree DG algebras

TL;DR

This paper proves that key equivalence problems for semifree noncommutative DG algebras are undecidable, resolving part of a known problem in low-dimensional topology.

Contribution

It establishes the undecidability of several fundamental isomorphism problems for semifree DG algebras, advancing understanding in algebraic topology.

Findings

Stable tame isomorphism problem is undecidable

Quasi-isomorphism problem is undecidable

Derived Morita equivalence problem is undecidable

Abstract

We prove that the stable tame isomorphism, quasi-isomorphism, and derived Morita equivalence problems for semifree noncommutative differential graded algebras (DGAs) are all undecidable. This resolves half of Problem 5.16 from the K3 Problem List in Low-Dimensional Topology. We present two solutions, both obtained (essentially autonomously) by Gemini Deep Think / Aletheia.