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arxiv: 2607.01889 · v1 · pith:6OBT3F5Cnew · submitted 2026-07-02 · 🧮 math.AT

Reduced characteristic number criteria for equivariant bordism of T^k- and (mathbb{Z}₂)^k-manifolds with isolated fixed points

Runze Chen , Zhi L\"u , Leqi Yang This is my paper

Pith reviewed 2026-07-03 02:19 UTC · model grok-4.3

classification 🧮 math.AT
keywords equivariant bordismT^k-manifolds(Z_2)^k-manifoldsisolated fixed pointsequivariant Chern classesStiefel-Whitney classesKosniowski's conjectureEuler characteristic
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The pith

A single polynomial of equivariant Chern classes determines equivariant bordism for unitary T^k-manifolds with isolated fixed points.

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

The paper develops reduced criteria that detect whether unitary T^k-manifolds and closed smooth (Z_2)^k-manifolds with isolated fixed points bound equivariantly, replacing the classical requirement of computing every equivariant characteristic number with far simpler data. For the T^k case the bordism class is captured by one polynomial in the equivariant Chern classes, and a minimal distinguishing degree supplies inequalities that relate manifold dimension to Euler characteristic. These inequalities establish the existence of a linear lower bound on the Euler characteristic inside the Kosniowski conjecture framework and supply an alternative proof of the toric case when dimension equals twice the torus rank. For (Z_2)^k-manifolds the criterion collapses further to powers of the top equivariant Stiefel-Whitney class. The resulting simplifications cut the computational cost of equivariant bordism checks substantially.

Core claim

For any unitary T^k-manifold M with isolated fixed points, an equivariant unitary bordism criterion is built entirely from a single polynomial of equivariant Chern classes. The minimal distinguishing degree produces two inequalities relating dim M and χ(M) that settle the existence of a linear lower bound for χ(M) in Kosniowski's conjecture and give an alternative proof of the toric generalization when dim M = 2k. For a closed smooth (Z_2)^k-manifold with isolated fixed points the equivariant bordism criterion relies solely on the powers of the top equivariant Stiefel-Whitney class.

What carries the argument

The single polynomial of equivariant Chern classes that encodes the reduced bordism criterion, together with the minimal distinguishing degree that governs the dimension-Euler characteristic relation.

If this is right

  • The full collection of equivariant characteristic numbers is unnecessary to decide equivariant bordism.
  • A linear lower bound on the Euler characteristic exists for these manifolds inside the Kosniowski framework.
  • The toric generalization of Kosniowski's conjecture holds when manifold dimension equals twice the torus rank.
  • Checking whether a (Z_2)^k-manifold bounds requires only powers of its top equivariant Stiefel-Whitney class.
  • Computational effort for equivariant bordism detection is substantially reduced.

Where Pith is reading between the lines

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

  • The reduced criteria could make explicit computation of higher-rank equivariant bordism groups feasible.
  • Similar polynomial reductions might be sought for actions whose fixed sets are not isolated.
  • The minimal distinguishing degree may supply new invariants in other equivariant cohomology theories.
  • Concrete examples such as toric varieties could now be checked algorithmically against the new criteria.

Load-bearing premise

The manifolds have isolated fixed points and satisfy the natural admissible assumptions of the Kosniowski conjecture framework.

What would settle it

A unitary T^k-manifold with isolated fixed points in which the single Chern-class polynomial vanishes yet the manifold fails to bound equivariantly, or in which the derived inequalities relating dimension and Euler characteristic are violated.

read the original abstract

Classical equivariant bordism theories require computing the full collection of equivariant characteristic numbers to detect whether an equivariant manifold bounds equivariantly or not. This paper establishes simplified equivariant bordism characterizations for two families of equivariant manifolds with isolated fixed points: unitary $T^k$-manifolds and closed smooth $(\mathbb{Z}_2)^k$-manifolds. For any unitary $T^k$-manifold $M$ with isolated fixed points, we establish an equivariant unitary bordism criterion built entirely from a single polynomial of equivariant Chern classes. We further introduce the minimal distinguishing degree and obtain two key inequalities that capture the interplay between $\dim M$ and the Euler characteristic $\chi(M)$ through this minimal distinguishing degree. These inequalities settle the existence problem of a linear lower bound for $\chi(M)$ within the framework of Kosniowski's conjecture and partially verify the conjecture under natural admissible assumptions. We also provide an alternative proof settling the toric generalization of Kosniowski's conjecture when $\dim M=2k$. By contrast, for a closed smooth $(\mathbb{Z}_2)^k$-manifold with isolated fixed points, we derive a more concise equivariant bordism criterion relying solely on the powers of the top equivariant Stiefel-Whitney class. Our new criteria substantially reduce computational demands.

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 claims to establish reduced equivariant bordism criteria for two classes of manifolds with isolated fixed points: for unitary T^k-manifolds, a single polynomial in the equivariant Chern classes determines whether the manifold bounds equivariantly; for closed smooth (Z_2)^k-manifolds, the criterion reduces to powers of the top equivariant Stiefel-Whitney class. The paper further defines a minimal distinguishing degree, derives two inequalities relating dim M and χ(M), and uses these to address Kosniowski's conjecture (including an alternative proof of its toric generalization when dim M = 2k) under stated admissible assumptions.

Significance. If the derivations hold, the results would meaningfully lower the computational burden in equivariant bordism by replacing the full set of characteristic numbers with a single polynomial or a short sequence of powers. The inequalities provide concrete progress on Kosniowski-type bounds and the alternative toric proof adds an independent verification route. These contributions are proportionate to the scope of the isolated-fixed-point setting and could be useful for explicit computations in low-dimensional cases.

minor comments (3)
  1. [Introduction] The abstract refers to 'natural admissible assumptions' for Kosniowski's conjecture; these should be stated explicitly (with references) in the introduction or in the section where the inequalities are proved so that the scope of the partial verification is immediately clear.
  2. [Main results] Notation for the single polynomial (e.g., its precise definition in terms of the equivariant Chern classes) and for the minimal distinguishing degree should be introduced with a numbered display equation at first use to facilitate later citations.
  3. [Examples] The manuscript would benefit from a short table or explicit low-dimensional example (e.g., dim M = 4 or 6) showing the reduction from the classical set of characteristic numbers to the new single-polynomial test.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our work on reduced equivariant bordism criteria for T^k- and (Z_2)^k-manifolds with isolated fixed points, the positive assessment of its significance in lowering computational demands, and the recommendation for minor revision. The referee's description accurately reflects the main results, including the single-polynomial criteria, the minimal distinguishing degree, the inequalities relating dimension and Euler characteristic, and the partial verification of Kosniowski's conjecture.

Circularity Check

0 steps flagged

No significant circularity; derivation rests on external classical bordism theory

full rationale

The paper's central claims establish reduced bordism criteria for manifolds with isolated fixed points using polynomials in equivariant Chern classes or powers of the top Stiefel-Whitney class. These are presented as consequences of classical equivariant bordism theory and the isolated-fixed-point hypothesis, without any visible self-definition of quantities, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the result to its own inputs. The Kosniowski inequalities and toric generalizations are derived conditionally on stated assumptions and external results, keeping the derivation self-contained against external benchmarks. No equations or constructions in the provided abstract or claim structure exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work is framed as a derivation within existing equivariant bordism theory.

pith-pipeline@v0.9.1-grok · 5781 in / 1091 out tokens · 33191 ms · 2026-07-03T02:19:11.775775+00:00 · methodology

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