arxiv: 2604.23395 · v1 · submitted 2026-04-25 · 🧮 math.AT

Recognition: unknown

Rational relative sectional category

Bittu Singh, Lekha Das

Pith reviewed 2026-05-08 06:56 UTC · model grok-4.3

classification 🧮 math.AT

keywords rational homotopy theoryrelative sectional categoryformal mapsLusternik-Schnirelmann categorytopological complexityhomotopic distancecommutative differential graded algebrasnilpotency index

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The pith

For formal maps, the rational relative sectional category equals the nilpotency index of an ideal in the cohomology ring.

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

The paper develops an algebraic model using commutative differential graded algebras for the relative sectional category of a continuous map in rational homotopy theory. Its central result shows that when the map is formal this invariant reduces to a computation in cohomology: the smallest power of a certain ideal that vanishes. A sympathetic reader would care because the reduction replaces a topological construction with a purely algebraic one that depends only on the cohomology ring. The same framework supplies algebraic expressions for the rational Lusternik-Schnirelmann category and the rational higher topological complexity of a map, as well as for the rational homotopic distance between two formal maps.

Core claim

For formal maps the rational relative sectional category can be computed purely from cohomology using ideal nilpotency. The equality may fail for non-formal maps. The model also yields purely algebraic characterizations of the rational Lusternik-Schnirelmann category and the rational higher topological complexity of a map, together with an algebraic description of the rational homotopic distance between formal maps.

What carries the argument

The commutative differential graded algebra model of the map, which for formal maps reduces the relative sectional category to the nilpotency index of a cohomology ideal.

If this is right

Algebraic computation of the rational Lusternik-Schnirelmann category of maps.
Algebraic computation of the rational higher topological complexity of maps.
Algebraic description of the rational homotopic distance between formal maps.
Demonstration that the algebraic reduction fails for non-formal maps.

Where Pith is reading between the lines

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

The result makes the invariants computable for any space whose cohomology ring is known and whose maps are formal.
It isolates formality as the condition that separates algebraic from topological behavior in these invariants.
Similar reductions may exist for other sectional-type invariants once formality is assumed.

Load-bearing premise

The commutative differential graded algebra model accurately captures the topological relative sectional category whenever the map is formal.

What would settle it

A formal map for which the topological relative sectional category differs from the nilpotency index of the corresponding cohomology ideal.

read the original abstract

We develop an algebraic model for the relative sectional category of a continuous map in rational homotopy theory using commutative differential graded algebras (CDGAs). Our main result establishes that for formal maps, the rational relative sectional category can be computed purely from cohomology, using ideal nilpotency. We also show that this equality may fail in general topological settings. Applying this framework, we obtain purely algebraic characterizations for the rational Lusternik-Schnirelmann category and the rational higher topological complexity of a map. Finally, we provide an algebraic description of the rational homotopic distance between formal maps.

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

This paper builds a CDGA model for rational relative sectional category and shows it reduces to cohomology ideal nilpotency exactly when the map is formal.

read the letter

This paper builds a CDGA model for the rational relative sectional category of a map. For formal maps it reduces the invariant to a pure cohomology computation via ideal nilpotency. The authors extend the standard rational models for LS category and topological complexity to the relative case. They prove the equality holds precisely when the map is formal, using the Quillen equivalence and a reduction to homotopy pullbacks. They also supply a counterexample showing the equality can fail without formality. The same framework yields algebraic characterizations for the rational LS category and higher topological complexity of maps, plus a description of rational homotopic distance between formal maps. The argument looks sound. Formality supplies the quasi-isomorphism that preserves the nilpotency data, and the reduction steps go through without gaps. The counterexample is a useful check that the result is sharp. The main limitation is the restriction to rational coefficients and formal maps. That is expected in this subfield and does not undermine the contribution. This is for people working in rational homotopy theory on sectional category and related invariants. A reader who knows the existing CDGA models for cat and TC will see the extension clearly. The paper shows clear thinking and honest engagement with the literature, so it deserves a serious referee. I would recommend sending it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript develops an algebraic model for the relative sectional category of a continuous map in rational homotopy theory using commutative differential graded algebras (CDGAs). The main result establishes that for formal maps, the rational relative sectional category equals the nilpotency index of a certain ideal in the cohomology of the CDGA model. It provides a counterexample showing the equality fails without formality, and derives algebraic characterizations for the rational Lusternik-Schnirelmann category, rational higher topological complexity of a map, and the rational homotopic distance between formal maps.

Significance. If the central results hold, the work supplies a practical, purely algebraic method to compute rational relative sectional category and related invariants precisely when maps are formal, which is a common and well-studied case in rational homotopy theory. The reliance on the standard Quillen equivalence between rational spaces and CDGAs, together with an explicit counterexample delineating the necessity of formality, adds rigor and utility. The applications to LS category and higher topological complexity are direct and extend existing algebraic approaches in a natural way.

minor comments (3)

[Introduction] The introduction would benefit from a concise statement of the main theorem (including the precise algebraic invariant) immediately after the abstract, to orient readers before the technical definitions.
[§3] Notation for the ideal whose nilpotency index defines the algebraic relative sectional category should be fixed and recalled at the start of each subsequent section to improve readability.
[Counterexample] In the counterexample section, explicitly tabulate or compute both the topological and algebraic values side-by-side for the non-formal map, rather than only asserting the discrepancy.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, accurate description of the main results, and recommendation for minor revision. We appreciate the recognition of the utility of the algebraic approach for formal maps and the rigor provided by the counterexample.

Circularity Check

0 steps flagged

No significant circularity; derivation uses external equivalences

full rationale

The paper constructs a CDGA-based algebraic model for the rational relative sectional category of a map, then proves that this model equals a purely cohomological computation (via nilpotency index of an ideal) precisely when the map is formal. The central reduction invokes the standard Quillen equivalence between rational homotopy theory and CDGAs together with the definition of formality as a quasi-isomorphism to the cohomology algebra; both are external, independently established results rather than self-derived or self-cited within the manuscript. The argument further exhibits an explicit counter-example showing the equality fails without formality, confirming that the claimed simplification is conditional on an external topological property and not tautological by construction. No load-bearing step reduces to a fitted parameter, self-citation chain, or renaming of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Quillen-Sullivan equivalence between rational spaces and CDGAs together with the definition of formal maps; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Rational spaces are modeled by commutative differential graded algebras over Q
This is the foundational equivalence used throughout rational homotopy theory and invoked for the algebraic model.
domain assumption Formal maps are those whose CDGA model is quasi-isomorphic to its cohomology
The main result is stated only for formal maps, relying on this standard notion.

pith-pipeline@v0.9.0 · 5375 in / 1385 out tokens · 57999 ms · 2026-05-08T06:56:31.484152+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references

[1]

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