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arxiv: 2607.00441 · v1 · pith:2GNBIQZInew · submitted 2026-07-01 · 🧮 math.SG

Tall Complexity One Spaces with k-colorable Skeleton

Yichen Liu This is my paper

Pith reviewed 2026-07-02 02:16 UTC · model grok-4.3

classification 🧮 math.SG
keywords tall complexity one spacesHamiltonian T-spacesskeletonone-skeletonk-colorableorbital moment mapsymplectic toric manifoldscomplexity one
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The pith

In tall complexity one Hamiltonian T-spaces, k-colorable skeletons have their information recovered from the one-skeleton.

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

Tall complexity one T-spaces are Hamiltonian T-spaces where half the dimension minus the torus dimension equals one, so symplectic quotients at each moment value are surfaces. The skeleton encodes non-generic orbits and acts as a central invariant for classification. The paper shows that if the skeleton admits a k-coloring, a partition into k closed and open subsets each carrying an injective orbital moment map, then the entire skeleton structure is determined by the one-skeleton alone. It further proves that any closed and open subset with injective orbital moment map can be realized exactly as the skeleton of a symplectic toric manifold obtained by extending the original T-action with an extra circle factor. This reduction focuses attention on low-dimensional orbit data rather than the full space.

Core claim

When the skeleton is k-colorable, i.e., when it can be partitioned into k closed and open subsets such that the orbital moment map is injective on each of them, its information can be recovered by the one-skeleton. For any closed and open subset of the skeleton on which the orbital moment map is injective, one can construct a symplectic toric (T×S¹)-manifold whose underlying complexity one T-space has the skeleton isomorphic to this subset.

What carries the argument

The k-colorability of the skeleton, defined as a partition into k closed and open subsets each having injective orbital moment map, which allows the full skeleton to be recovered from the one-skeleton of non-generic orbits of dimension at most one.

If this is right

  • Classification of these spaces reduces to combinatorial or topological properties of the one-skeleton under the colorability hypothesis.
  • Every injective closed-open subset of a skeleton arises from some explicit toric extension of the T-action.
  • The non-generic orbit data is fully captured by the orbits of dimension at most one once the partition exists.
  • The surface nature of the symplectic quotients makes the skeleton amenable to partition-based analysis.

Where Pith is reading between the lines

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

  • The colorability condition may turn the geometric classification problem into a question about graph or complex colorings on the orbit space.
  • The toric extension construction supplies a systematic way to produce examples with any prescribed injective subset as skeleton.
  • Similar partition techniques could be tested on complexity-one spaces that are not tall or on spaces with higher complexity.
  • One could look for whether every tall complexity one space admits at least a finite coloring of this type.

Load-bearing premise

The spaces are compact and connected tall complexity one T-spaces, and the skeleton is assumed to admit the stated k-coloring with injective orbital moment maps on each part.

What would settle it

A counterexample would be a compact connected tall complexity one T-space whose k-colorable skeleton cannot have its full structure recovered from the one-skeleton, or a closed-open injective subset that cannot be realized as the skeleton of any symplectic toric (T×S¹)-manifold.

read the original abstract

Tall complexity one $T$-spaces are Hamiltonian $T$-spaces $(M,\omega,\Phi)$ such that $\frac{1}{2}\dim M -\dim T=1$ and the symplectic quotient at each moment value is a surface. The skeleton of a complexity one $T$-space is an important invariant in the classification and encodes the information about non-generic orbits. In this paper, we study properties of the skeleton of a compact, connected tall complexity one $T$-spaces. We prove that when the skeleton is $k$-colorable, i.e., when it can be partitioned into $k$ closed and open subsets such that the orbital moment map is injective on each of them, its information can be recovered by the one-skeleton (the set of non-generic orbits whose dimension is at most one). We also prove that for any cloesd and open subset of the skeleton on which the orbital moment map is injective, one can construct a symplectic toric $(T\times S^1)$-manifold whose underlying complexity one $T$-space has the skeleton isomorphic to this subset.

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 / 2 minor

Summary. The paper examines compact, connected tall complexity one T-spaces (Hamiltonian T-spaces with ½ dim M − dim T = 1 and surface symplectic quotients). It proves that if the skeleton is k-colorable (partitionable into k closed-and-open subsets on which the orbital moment map is injective), then the full skeleton information is recoverable from the one-skeleton. It further proves that any closed-and-open subset of the skeleton on which the orbital moment map is injective arises as the skeleton of the complexity-one T-space underlying some symplectic toric (T × S¹)-manifold.

Significance. If the stated theorems hold, the results supply a reconstruction theorem for the skeleton from its one-skeleton under an explicit colorability hypothesis and a realization theorem for admissible subsets. Both statements are conditional on the given hypotheses and therefore constitute targeted advances within the existing framework for classifying Hamiltonian torus actions of complexity one.

minor comments (2)
  1. [Abstract] Abstract, line beginning 'We also prove that for any cloesd': 'cloesd' is a typographical error for 'closed'.
  2. [Abstract] Abstract: the sentence 'its information can be recovered by the one-skeleton' would benefit from a brief parenthetical gloss on what 'information' is being recovered (e.g., the combinatorial type or the moment-map image).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its significance within the classification of complexity-one Hamiltonian torus actions, and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No circularity; theorems conditional on explicit hypotheses with no self-referential reductions

full rationale

The paper consists of classification theorems in symplectic geometry for tall complexity-one T-spaces. All central claims are explicitly conditional on the skeleton being k-colorable (partitioned into closed/open subsets with injective orbital moment map) and on the spaces being compact, connected, and tall. The abstract and described results contain no equations, fitted parameters, predictions derived from subsets of data, or load-bearing self-citations. The one-skeleton recovery and construction statements are proved under these stated hypotheses rather than reducing to them by definition or ansatz. No step equates an output to an input by construction, and the derivation chain is self-contained against external mathematical benchmarks in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no explicit free parameters, invented entities, or ad-hoc axioms listed. Relies on standard domain assumptions in symplectic geometry such as definitions of Hamiltonian T-spaces and moment maps.

axioms (1)
  • domain assumption Standard definitions and properties of Hamiltonian T-spaces, complexity one spaces, and the orbital moment map hold as background.
    The claims presuppose the established theory of symplectic quotients and skeletons without re-deriving them.

pith-pipeline@v0.9.1-grok · 5712 in / 1315 out tokens · 24179 ms · 2026-07-02T02:16:15.159472+00:00 · methodology

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Reference graph

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