arxiv: 2605.08122 · v1 · submitted 2026-04-28 · 🧮 math.RA · math.GT· math.LO· math.SG

Recognition: no theorem link

Undecidability problems for semifree DG algebras

Ciprian Manolescu, Nick Rozenblyum

Pith reviewed 2026-05-12 01:01 UTC · model grok-4.3

classification 🧮 math.RA math.GTmath.LOmath.SG

keywords undecidabilitydifferential graded algebrassemifree DGAsstable tame isomorphismquasi-isomorphismderived Morita equivalencenoncommutative algebra

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The pith

The stable tame isomorphism, quasi-isomorphism, and derived Morita equivalence problems for semifree noncommutative differential graded algebras are all undecidable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that three core decision problems for semifree noncommutative differential graded algebras cannot be solved by any algorithm. These problems concern whether two such algebras are related by stable tame isomorphism, by quasi-isomorphism, or by derived Morita equivalence. A reader would care because the results show that algebraic classifications used in low-dimensional topology are computationally intractable in general, settling one direction of an open question in the area.

Core claim

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.

What carries the argument

Explicit constructions of semifree DGAs together with reductions from known undecidable problems that preserve the relevant equivalence relations.

If this is right

No algorithm exists that can determine whether two arbitrary semifree DGAs are stably tamely isomorphic.
The quasi-isomorphism relation between semifree DGAs is not decidable by any computational procedure.
Derived Morita equivalence of semifree DGAs cannot be decided algorithmically in general.
Computational methods in noncommutative algebra and related topological invariants must work around these undecidability barriers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same style of reduction may apply to other equivalence relations on DGAs that appear in topological invariants.
Researchers could usefully identify natural subclasses of semifree DGAs on which one or more of these problems become decidable.
The undecidability results suggest exploring whether similar hardness holds for commutative or other restricted classes of differential graded algebras.

Load-bearing premise

The explicit constructions and reductions from known undecidable problems to the three DGA equivalence problems are valid and preserve the relevant algebraic structures without introducing decidable special cases.

What would settle it

An algorithm that decides stable tame isomorphism for arbitrary semifree noncommutative DGAs, or a concrete pair of semifree DGAs for which one of the reductions maps an undecidable instance to a decidable one.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows undecidability for stable tame isomorphism, quasi-isomorphism, and derived Morita equivalence on semifree DGAs via two explicit reductions from known undecidable problems, resolving half of an open question.

read the letter

This paper proves that three equivalence problems for semifree noncommutative DG algebras are undecidable. It does so by reducing from the word problem in finitely presented groups and from matrix semigroup mortality, with the constructions laid out in sections 3 and 4. The resulting DGAs stay semifree, and the equivalences are preserved exactly when the original instance holds, checked by direct computation of the relevant maps and homotopies. No obvious special cases that would make the problems decidable slip in. That is the core contribution, and it directly addresses half of Problem 5.16 on the K3 list. The two solutions are presented as independent, which adds some robustness even if both trace back to the same AI-assisted process. The reductions themselves look parameter-free and grounded in standard computability facts, so the argument does not rest on circular or self-referential steps. One minor soft spot is that the paper only claims to resolve half the listed problem, leaving the other half open, but that is stated clearly rather than overstated. The work is aimed at people who care about decidability questions at the algebra-topology boundary, especially those already familiar with DGAs and the K3 list. A reader in that niche gets a concrete negative result with explicit constructions to examine. I would send this to peer review; the reductions are the kind of thing referees can check directly, and the claim is narrow enough to be verifiable.

Referee Report

0 major / 0 minor

Summary. The manuscript proves that the stable tame isomorphism, quasi-isomorphism, and derived Morita equivalence problems for semifree noncommutative differential graded algebras (DGAs) are all undecidable. It presents two solutions obtained by reductions from the word problem for finitely presented groups and from the matrix semigroup mortality problem. The constructions in §3 and §4 map instances of these problems to semifree DGAs such that the algebraic equivalences hold if and only if the original instance does.

Significance. Assuming the validity of the reductions and verifications, the result is significant for establishing undecidability in these classification problems within the category of semifree DGAs. This has implications for computational algebra and resolves part of an open problem in low-dimensional topology. The explicit nature of the constructions and the direct computations of chain maps and homotopies to show preservation of equivalences are positive aspects of the work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We appreciate the recognition of the significance of the undecidability results for stable tame isomorphism, quasi-isomorphism, and derived Morita equivalence in the category of semifree noncommutative DGAs, as well as the implications for computational algebra and low-dimensional topology.

Circularity Check

0 steps flagged

No significant circularity; result obtained by explicit reduction from independent undecidable problems

full rationale

The paper establishes undecidability of the three DGA problems via two explicit reductions (detailed in §§3–4) from the word problem for finitely presented groups and from matrix semigroup mortality. These constructions map instances directly to semifree DGAs while preserving the equivalence relations by direct verification of chain maps and homotopies; no parameter fitting, self-definition, or load-bearing self-citation is used. The grounding is external (standard computability results) and the manuscript is self-contained against those benchmarks, with no step reducing by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard results from algebra, differential graded algebras, and computability theory together with explicit reductions; no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)

standard math Standard axioms of set theory, algebra, and computability theory
Invoked implicitly to support the existence of reductions from known undecidable problems.

pith-pipeline@v0.9.0 · 5356 in / 1149 out tokens · 110542 ms · 2026-05-12T01:01:56.091841+00:00 · methodology

discussion (0)

Reference graph

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