A commutative model for PL compactly supported cohomology in characteristic zero

Tom Sutton

arxiv: 1907.04454 · v1 · pith:6FLHCCXFnew · submitted 2019-07-09 · 🧮 math.AT

A commutative model for PL compactly supported cohomology in characteristic zero

Tom Sutton This is my paper

Pith reviewed 2026-05-24 23:36 UTC · model grok-4.3

classification 🧮 math.AT

keywords rational homotopy theorycompactly supported cohomologyde Rham theoremSullivan modelPL topologycharacteristic zeroproper homotopy theory

0 comments

The pith

Sullivan rational homotopy theory extends to the compactly supported setting with a simplicial de Rham model.

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

The paper extends classical rational homotopy theory to handle spaces with compact supports. It constructs a simplicial model based on the compactly supported de Rham complex in characteristic zero. This model is shown to compute the singular compactly supported cohomology. A sympathetic reader would care because it allows algebraic study of proper homotopy types for non-compact spaces.

Core claim

We show that classical rational homotopy theory in the sense of Sullivan can be extended to the compactly supported setting. This presents a simplicial version of the compactly supported de Rham complex in characteristic zero, proving that it models singular compactly supported cohomology.

What carries the argument

the simplicial version of the compactly supported de Rham complex, which serves as a commutative model for PL compactly supported cohomology

If this is right

It presents new avenues of possible study in proper homotopy theory.
It allows further extensions of the ideas within the classical literature of Quillen and Sullivan.
The model works in characteristic zero for PL spaces.

Where Pith is reading between the lines

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

This could enable computations of compactly supported cohomology groups using differential graded algebras for non-compact manifolds.
Similar extensions might apply to other variants of homotopy theory like equivariant or parametrized versions.
Testing on examples like Euclidean spaces could verify the model's accuracy.

Load-bearing premise

The classical constructions and properties from Sullivan's rational homotopy theory extend to the compactly supported case with only the modifications needed to incorporate compact supports, without additional technical obstructions.

What would settle it

Finding a non-compact PL manifold where the simplicial compactly supported de Rham complex is not quasi-isomorphic to the singular cochain complex with compact supports.

read the original abstract

We show that that classical rational homotopy theory in the sense of Sullivan [6] can be extended compactly supported setting. This presents a simplicial version of the compactly supported de Rham complex in characteristic zero, and proving that it models singular compactly supported cohomology. This presents new avenues of possible study in proper homotopy theory, and further extensions of the ideas within the classical literature of Quillen [5] and Sullivan.

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

Claims a simplicial model extending Sullivan to compactly supported cohomology but the abstract supplies no construction or proof details.

read the letter

This paper claims to extend Sullivan's rational homotopy theory to the compactly supported setting by producing a simplicial version of the compactly supported de Rham complex in characteristic zero that models singular compactly supported cohomology. That is the central new claim, and it targets a genuine gap for people working with non-compact spaces or proper maps. If the construction works, it would let rational models apply more directly to proper homotopy questions, which is a reasonable direction to pursue from the Quillen-Sullivan literature. The abstract states the modeling property cleanly and notes the potential for further extensions, which is fair enough as far as it goes. The main soft spot is the complete absence of any construction details, proof outline, or verification steps in the abstract. Without seeing how the simplicial object is actually defined or how the modeling isomorphism is established, it is impossible to check whether the classical properties really carry over with only the obvious changes for compact supports or whether new obstructions appear. The assumption that the extension is straightforward therefore remains untested from the given text. This work is aimed at algebraic topologists already comfortable with Sullivan models who want to handle compact supports or proper maps. A reader in that niche could get value from the model once the details are checked. It deserves a serious referee because the claim is specific enough to be worth evaluating the actual construction and proof rather than desk-rejecting on the abstract alone. Recommendation: send it to peer review so the details can be examined.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to extend classical rational homotopy theory in the sense of Sullivan to the compactly supported setting. It presents a simplicial version of the compactly supported de Rham complex in characteristic zero and asserts that this models singular compactly supported cohomology, opening avenues in proper homotopy theory.

Significance. If the claimed construction and modeling property hold with the necessary proofs, the work would extend Sullivan's and Quillen's frameworks to proper homotopy theory. However, the provided abstract supplies no construction, definitions, or proof outline, so the potential significance cannot be evaluated from the given material.

major comments (1)

The abstract asserts the existence of the model and the modeling property but supplies no construction details, proof outline, or verification steps, making it impossible to determine whether the mathematics supports the central claim.

minor comments (1)

The abstract contains grammatical issues, including the repeated 'that' in the first sentence and the incomplete phrasing 'and proving that it models' in the second sentence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments. While the abstract is concise, the full manuscript contains the detailed construction and proofs; we will revise the abstract to better outline these elements.

read point-by-point responses

Referee: The abstract asserts the existence of the model and the modeling property but supplies no construction details, proof outline, or verification steps, making it impossible to determine whether the mathematics supports the central claim.

Authors: The full manuscript constructs a simplicial commutative model for the PL compactly supported de Rham complex in characteristic zero and proves that it models singular compactly supported cohomology, thereby extending Sullivan's rational homotopy theory to the compactly supported setting. We agree the abstract is too brief and lacks an outline of the construction or key verification steps. We will revise the abstract to include a concise description of the model and the modeling property. revision: yes

Circularity Check

0 steps flagged

No circularity identified; derivation chain not visible in provided text

full rationale

The abstract and context reference an extension of Sullivan's rational homotopy theory to the compactly supported case but supply no equations, definitions, proofs, or self-citations. Without the full manuscript's derivation steps, no load-bearing reduction to inputs by construction, fitted prediction, or self-citation chain can be exhibited. The paper appears self-contained against external benchmarks (Sullivan [6], Quillen [5]) with no visible circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal implied assumptions needed for the claimed extension to be possible. No explicit free parameters, invented entities, or detailed axioms are stated.

axioms (1)

domain assumption Sullivan's rational homotopy theory admits a direct extension to the compactly supported setting that preserves the modeling property for cohomology.
The abstract states that the classical theory can be extended and that the resulting model works, so this background assumption is required for the claim.

pith-pipeline@v0.9.0 · 5578 in / 1185 out tokens · 24815 ms · 2026-05-24T23:36:15.036027+00:00 · methodology

A commutative model for PL compactly supported cohomology in characteristic zero

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)