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arxiv: 2509.16759 · v4 · submitted 2025-09-20 · 🧮 math.GT · math.AT· math.GR

On distributional topological complexity of groups and manifolds

Alexander Dranishnikov This is my paper

Pith reviewed 2026-05-18 15:32 UTC · model grok-4.3

classification 🧮 math.GT math.ATmath.GR
keywords distributional topological complexitytopological complexitydistributional LS-categorylens spaceshyperbolic groupsnilpotent groupsproduct formulacounterexamples
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The pith

Distributional topological complexity equals the standard version for torsion-free hyperbolic and nilpotent groups.

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

The paper proves that distributional topological complexity equals ordinary topological complexity for torsion-free hyperbolic groups and for torsion-free nilpotent groups. It also shows that lens spaces satisfy explicit upper bounds on distributional topological complexity and on distributional LS-category, with those bounds becoming sharp when the order p is prime and the dimension n is large enough. These bounds are applied directly to construct counterexamples showing that the product formula fails for both distributional LS-category and distributional topological complexity. A sympathetic reader cares because the results clarify when variant complexity measures coincide on groups and reveal concrete cases where multiplicative expectations break for manifolds.

Core claim

For torsion-free hyperbolic groups Γ and torsion-free nilpotent groups Γ the equality dTC(Γ) = TC(Γ) holds. For the lens space L^n_p the inequalities dTC(L^n_p) ≤ 2p-1 and dcat(L^n_p) ≤ p-1 are proved, and both become equalities when p is prime and n > p. These inequalities are used to produce counterexamples to the product formula for dcat and for dTC.

What carries the argument

The distributional topological complexity dTC, a variant of ordinary topological complexity TC, together with the distributional LS-category dcat, applied to groups and to lens spaces to test equalities and product formulas.

If this is right

  • dTC coincides with TC on all torsion-free hyperbolic groups.
  • dTC coincides with TC on all torsion-free nilpotent groups.
  • Lens spaces L^n_p obey the concrete bound dTC ≤ 2p-1.
  • Lens spaces L^n_p obey the concrete bound dcat ≤ p-1, which is sharp for prime p and n > p.
  • The product formula for dcat and for dTC fails on certain lens spaces.

Where Pith is reading between the lines

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

  • The equality dTC = TC might extend to other torsion-free groups such as free groups or fundamental groups of closed hyperbolic manifolds.
  • The lens-space bounds suggest similar estimates could hold for other spherical space forms or manifolds with cyclic fundamental group.
  • The counterexamples indicate that any general product formula for these invariants must impose extra conditions such as simple connectedness.

Load-bearing premise

The groups must be torsion-free hyperbolic or torsion-free nilpotent and the lens spaces must be of the stated form L^n_p with the given conditions on p and n.

What would settle it

An explicit example of a torsion-free hyperbolic group for which dTC differs from TC, or a lens space L^n_p where the stated upper bound on dTC is violated.

read the original abstract

We prove the equality $\dTC(\Gamma)=\TC(\Gamma)$ for distributional topological complexity of torsion free hyperbolic and of torsion free nilpotent groups. For the distributional topological complexity of lens spaces we prove the inequality $\dTC(L^n_p)\le 2p-1$ and for the distributional LS-category the inequality $d\cat(L^n_p)\le p-1$ which turns into equality for prime $p$ and $n>p$. We use these inequalities to bring counter-examples to the product formula for $d\cat$ and $\dTC$.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that dTC(Γ) equals TC(Γ) for torsion-free hyperbolic groups and torsion-free nilpotent groups Γ. For lens spaces L^n_p it establishes the inequalities dTC(L^n_p) ≤ 2p−1 and dcat(L^n_p) ≤ p−1, with the latter becoming an equality when p is prime and n > p. These bounds are applied to produce counterexamples to product formulas for dcat and dTC.

Significance. If the derivations hold, the work supplies explicit equalities and bounds for distributional topological complexity and LS-category on concrete classes of groups and manifolds. The counterexamples to product formulas are a useful contribution, as they separate the distributional invariants from their classical counterparts using standard definitions and prior results from the literature.

minor comments (3)
  1. [Abstract] Abstract: the statement of the equality dTC(Γ)=TC(Γ) would benefit from a brief parenthetical reference to the precise definitions of dTC and TC employed (e.g., citing the relevant earlier papers).
  2. [Introduction] The notation L^n_p for lens spaces should be defined explicitly in the introduction, including the range of n and p under consideration and the standard action used to form the quotient.
  3. [Section on counterexamples] When the inequalities are turned into counterexamples to product formulas, a short table or explicit numerical example (e.g., specific p, n, and product manifold) would make the failure of the formula immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our results on distributional topological complexity for groups and manifolds, the significance assessment, and the recommendation of minor revision. We are pleased that the explicit equalities, bounds, and counterexamples to product formulas are viewed as a useful contribution.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper proves dTC(Γ)=TC(Γ) for torsion-free hyperbolic and nilpotent groups, plus inequalities dTC(L^n_p)≤2p-1 and dcat(L^n_p)≤p-1 (with equality cases), by applying standard external definitions of TC, dTC, and dcat together with group-theoretic and topological properties from the literature. These steps do not reduce to self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations; the counterexamples to product formulas are direct consequences of the derived inequalities rather than circular inputs. The argument chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions and background results from algebraic topology and geometric group theory; no free parameters, new entities, or ad-hoc axioms are introduced in the stated claims.

axioms (2)
  • domain assumption Standard definitions of TC, dTC, cat, and dcat from prior literature in algebraic topology.
    Invoked throughout to state the equalities and inequalities.
  • domain assumption Known structural properties of torsion-free hyperbolic and nilpotent groups.
    Used to establish the equality dTC(Γ)=TC(Γ).

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