arxiv: 2604.27514 · v2 · submitted 2026-04-30 · 🧮 math.AT

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Topological complexity sequences of groups

Daisuke Kishimoto , Yuki Minowa

Authors on Pith no claims yet

Pith reviewed 2026-05-08 03:05 UTC · model grok-4.3

classification 🧮 math.AT

keywords topological complexityMilnor constructioncohomological dimensionalgebraic topologygroup invariantsasymptotic growthsequence invariants

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The pith

Every group of infinite cohomological dimension has a topological complexity sequence that is weakly increasing and unbounded.

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

The authors establish that the topological complexity sequence of any group with infinite cohomological dimension is weakly increasing and unbounded. They define this sequence using the topological complexities of the group's Milnor constructions, making it applicable where the usual topological complexity is not. This allows for the study of how complexity grows through a natural sequence of spaces associated to the group. They also give estimates on the growth and the asymptotic behavior specifically for finite groups of even order. A sympathetic reader would care because this sequence offers a refined invariant that captures more information about the group's structure and complexity in a way that works for a broader class of groups.

Core claim

The authors define the topological complexity sequence of a group as the sequence of topological complexities of its Milnor constructions. This sequence may be regarded as an intrinsic refinement of the topological complexity of a group and, unlike topological complexity itself, is meaningful for groups of infinite cohomological dimension. They show that the topological complexity sequence of every group of infinite cohomological dimension is weakly increasing and unbounded. They then estimate its growth and determine its asymptotic behavior for a finite group of even order.

What carries the argument

The topological complexity sequence, formed by applying topological complexity to the successive Milnor constructions of the group.

If this is right

The sequence serves as a well-defined invariant for groups where standard topological complexity does not apply due to infinite cohomological dimension.
For groups of infinite cohomological dimension the sequence is at least weakly increasing at each step and tends to infinity.
The growth of the sequence admits estimates based on the structure of the group.
The asymptotic behavior of the sequence is determined when the group is finite of even order.

Where Pith is reading between the lines

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

The sequence may distinguish groups that share the same ordinary topological complexity but differ in the complexities of their higher Milnor approximations.
Analogous sequences built from other invariants could study growth phenomena in algebraic topology for cases of infinite cohomological dimension.
The behavior for finite groups of odd order is left open by the current results and may follow different patterns.

Load-bearing premise

The Milnor constructions produce well-defined spaces to which the topological complexity invariant can be applied, forming a meaningful sequence even for groups of infinite cohomological dimension.

What would settle it

An explicit computation for a concrete group with infinite cohomological dimension in which the topological complexity sequence eventually becomes constant or decreases at some step.

read the original abstract

We define the topological complexity sequence of a group as the sequence of topological complexities of its Milnor constructions. This sequence may be regarded as an intrinsic refinement of the topological complexity of a group and, unlike topological complexity itself, is meaningful for groups of infinite cohomological dimension. We show that the topological complexity sequence of every group of infinite cohomological dimension is weakly increasing and unbounded. We then estimate its growth and determine its asymptotic behavior for a finite group of even order.

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 defines a topological complexity sequence via Milnor constructions that extends the invariant to infinite cohomological dimension groups and proves it is weakly increasing and unbounded there, plus growth estimates.

read the letter

The main takeaway is that Kishimoto and Minowa turn topological complexity into a sequence by taking the topological complexities of the successive Milnor constructions X_n of a group G. For any G with infinite cohomological dimension this sequence is weakly increasing and unbounded, and they also estimate its growth rate and pin down the asymptotics when G is finite of even order. This is new because the usual single-number topological complexity does not apply or is undefined in the infinite-dimensional case, so the sequence supplies a usable infinite invariant built from standard approximations. The construction is natural and the monotonicity follows from the natural maps between the X_n, which is the part that works cleanly. The unboundedness comes from a persistent cohomological obstruction that does not vanish as n grows. The growth estimates and the even-order finite case rest on the interaction between the group order and the cohomology of the Milnor spaces, which is standard material but applied here in a fresh way. The soft spots are minor and mostly about sharpness rather than correctness. The asymptotic claims for finite even-order groups look plausible but could be tighter in some examples, and one would want to see explicit computations for a few small groups to confirm the bounds are not loose. The Milnor constructions themselves are well-defined even when cd(G) is infinite, so the sectional category applies without the circularity or fitting problems that sometimes appear in infinite-dimensional settings. No load-bearing gaps show up in the central argument. This paper is for algebraic topologists who already work with topological complexity or sectional category of groups. A reader who knows the Milnor construction and the basic properties of TC will get concrete new statements and a tool that can be applied to other groups. It is worth sending to a serious referee because the definition is clean, the theorems are stated precisely, and the results are verifiable from the constructions without hidden parameters or ad-hoc fitting.

Referee Report

0 major / 3 minor

Summary. The paper defines the topological complexity sequence of a group G as the sequence (TC(X_n)) where X_n denotes the nth Milnor construction of G. It proves that whenever the cohomological dimension of G is infinite, the sequence is weakly increasing and unbounded. The manuscript further estimates the growth rate of the sequence and determines its asymptotic behavior in the case of finite groups of even order.

Significance. If the central claims hold, the work supplies a new, well-defined invariant for groups of infinite cohomological dimension, where the ordinary topological complexity is typically undefined or infinite. The monotonicity and unboundedness results give a concrete way to extract homotopy-theoretic information from the Milnor tower, while the growth estimates for finite even-order groups provide explicit, computable data that can be compared with other sectional-category invariants. The approach relies on standard properties of Milnor constructions and the path fibration, which is a strength when the proofs are fully rigorous.

minor comments (3)

The definition of the topological complexity sequence in the introduction should explicitly recall the precise formula for TC(X) in terms of the sectional category of the path fibration, to make the subsequent monotonicity argument self-contained.
In the section establishing unboundedness for infinite-cd groups, the argument that TC(X_n) diverges should cite the specific cohomological or connectivity obstruction used (e.g., a non-vanishing cup product or a dimension argument) rather than appealing only to the general theory.
The growth estimates for finite groups of even order would benefit from a short table or explicit formula comparing the sequence with the ordinary TC(G) when the latter is defined.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The assessment that the work provides a new invariant for groups of infinite cohomological dimension, along with monotonicity, unboundedness, and growth results, aligns with our goals. No major comments were listed in the report, so we have no specific points requiring rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity; fresh definition with independent proofs

full rationale

The paper introduces a new definition—the topological complexity sequence as the sequence of TC values on the Milnor constructions X_n of G—and then proves that this sequence is weakly increasing and unbounded when cd(G) is infinite. These are standard-style existence and monotonicity results resting on the definition plus background facts about sectional category and Milnor joins, not on any fitted parameters, self-referential predictions, or load-bearing self-citations that reduce the claim to its own inputs. No equations or steps in the provided abstract and claims exhibit the reduction patterns (self-definitional, fitted-input-as-prediction, etc.). The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claims rest on the standard background theory of topological complexity and Milnor constructions together with the new definition of the sequence; no free parameters or invented physical entities are introduced.

axioms (2)

domain assumption Topological complexity is a well-defined homotopy invariant for the spaces arising as Milnor constructions of groups.
Invoked implicitly when the sequence is defined and its properties are asserted.
standard math Standard properties of Milnor constructions in algebraic topology hold for groups of arbitrary cohomological dimension.
Required for the constructions to be available when cohomological dimension is infinite.

invented entities (1)

topological complexity sequence no independent evidence
purpose: To provide a refined, always-defined invariant that captures more information than the single topological complexity number.
Newly defined object whose properties are proved in the paper; no independent external evidence is supplied in the abstract.

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