Higher Homotopic Distance

Ayse Borat; Tane Vergili

arxiv: 1907.08384 · v1 · pith:JVK67BWVnew · submitted 2019-07-19 · 🧮 math.AT

Higher Homotopic Distance

Ayse Borat , Tane Vergili This is my paper

Pith reviewed 2026-05-24 19:10 UTC · model grok-4.3

classification 🧮 math.AT

keywords higher homotopic distanceLusternik-Schnirelmann categorysectional categorytopological complexityhigher topological complexityhomotopy invariantsalgebraic topology

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The pith

Higher homotopic distance equals cat, secat and higher topological complexity under identified conditions.

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

The paper introduces several properties of the higher homotopic distance. It determines the conditions under which this distance coincides with the Lusternik-Schnirelmann category, the sectional category, and higher-dimensional topological complexity. It supplies alternative proofs for certain theorems about higher topological complexity by relying on the higher homotopic distance instead of prior methods.

Core claim

The higher homotopic distance, as introduced in a cited reference, possesses properties that make it equal to cat, secat, and TC_n precisely when the paper's stated conditions hold, and these equalities in turn furnish alternative proofs for some TC_n-related theorems.

What carries the argument

Higher homotopic distance, the higher-dimensional analog of homotopic distance whose equality to cat, secat and TC_n is characterized under explicit conditions.

If this is right

Equality holds between higher homotopic distance and cat, secat, and TC_n when the paper's conditions are met.
Some existing theorems on TC_n admit alternative proofs that route through higher homotopic distance.
Several basic properties of higher homotopic distance are established that support the equality comparisons.

Where Pith is reading between the lines

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

The equalities may allow known bounds or computations for cat to be transferred directly to higher homotopic distance.
The alternative proofs could be applied to obtain new results for other spaces where TC_n is studied.
Similar unification via higher homotopic distance might extend to additional higher invariants in algebraic topology.

Load-bearing premise

The higher homotopic distance is a well-defined invariant that can be directly compared to cat, secat, and higher TC_n via the stated conditions.

What would settle it

A concrete space satisfying the paper's equality conditions yet where the numerical value of higher homotopic distance differs from that of cat, secat or TC_n.

read the original abstract

The concept of homotopic distance and its higher analog are introduced in [6]. In this paper we introduce some important properties of higher homotopic distance, investigate the conditions under which $\cat$, $\secat$ and higher dimensional topological complexity are equal to the higher homotopic distance, and give alternative proofs, using higher homotopic distance, to some $\TC_n$-related theorems.

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

A direct follow-up that adds properties and equality conditions to the higher homotopic distance from [6], plus alternative proofs for some TC_n results.

read the letter

The main takeaway is that this paper works out properties of the higher homotopic distance introduced in the prior reference and spells out conditions under which it equals cat, secat, and higher TC_n. It also supplies alternative proofs for a few existing TC_n theorems using this invariant. That is the actual content. It does the alternative proofs part cleanly enough on its own terms, which can be handy if someone is already computing with these quantities and wants another route. The equality conditions are stated explicitly, so they can be checked against examples. The soft spots are that everything rests on the definition from [6] without re-examining it, and the contribution stays inside that same program rather than producing new computations or broader applications. No load-bearing gaps show up in the abstract, but the advance is incremental by design. This is for readers already working on topological complexity invariants who need the relations or the alternative arguments for their own proofs. It is not aimed at outsiders or at reorganizing the area. The claims are specific enough to referee, so I would send it out rather than desk reject.

Referee Report

0 major / 2 minor

Summary. The manuscript builds on the definition of higher homotopic distance from the cited reference [6]. It derives several properties of this invariant, identifies conditions under which it equals the Lusternik-Schnirelmann category (cat), sectional category (secat), and higher topological complexity (TC_n), and supplies alternative proofs for selected TC_n-related theorems.

Significance. If the equality conditions and alternative proofs are valid, the work supplies a unifying perspective on these invariants and may streamline arguments in the theory of topological complexity. The explicit characterization of when the invariants coincide is a concrete contribution that could be used in subsequent comparisons.

minor comments (2)

The abstract refers to 'some important properties' and 'conditions under which' equality holds without naming the principal results; a sentence listing the main theorems or the key equality criteria would improve readability.
Reference [6] is the foundational source for the definition; the manuscript should include a complete bibliographic entry (journal, volume, year, pages) rather than leaving it as a placeholder citation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the accurate summary of its contributions and the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper takes the definition of higher homotopic distance as given from the external reference [6] and proceeds to derive new properties, equality conditions with cat/secat/TC_n, and alternative proofs. No step reduces by the paper's own equations to a fitted input, self-definition, or unverified self-citation chain; the cited definition functions as an independent starting point, and the new results are obtained via direct comparison and proof under explicitly stated conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates entirely within standard homotopy theory; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)

standard math Standard axioms and constructions of algebraic topology and homotopy theory (spaces, maps, homotopies, fibrations).
All statements presuppose the usual category of topological spaces and the homotopy category.

pith-pipeline@v0.9.0 · 5568 in / 1077 out tokens · 21881 ms · 2026-05-24T19:10:25.704302+00:00 · methodology

Higher Homotopic Distance

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)