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arxiv: 2511.10153 · v4 · submitted 2025-11-13 · 🧮 math.GT · math.AG

Classification of locally standard T-pseudomanifolds over topological stratified pseudomanifolds

Yuya Koike , Shintaro Kuroki This is my paper

Pith reviewed 2026-05-17 22:20 UTC · model grok-4.3

classification 🧮 math.GT math.AG
keywords locally standard T-pseudomanifoldstopological stratified pseudomanifoldscharacteristic dataequivariant homeomorphismquasitoric manifoldstoric varietiesclassification theorem
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The pith

Locally standard T-pseudomanifolds over topological stratified pseudomanifolds are completely classified up to weakly equivariant homeomorphism by their characteristic data.

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

The paper introduces locally standard T-pseudomanifolds as objects that generalize complete toric varieties and locally standard T-manifolds. It proves that when the base is a topological stratified pseudomanifold meeting certain conditions, these structures are fully determined up to weakly equivariant homeomorphism by their characteristic data. This extends the earlier classification of quasitoric manifolds. A reader would care because the result supplies a complete discrete invariant for a wider class of spaces with torus symmetries, including singular ones. The work unifies classification results across smooth and stratified settings.

Core claim

Locally standard T-pseudomanifolds over topological stratified pseudomanifolds satisfying certain conditions are completely classified, up to (weakly) equivariant homeomorphism, by their characteristic data. This result extends the classification of quasitoric manifolds by Davis-Januszkiewicz.

What carries the argument

Characteristic data, the combinatorial and topological invariant that determines the equivariant homeomorphism type of the locally standard T-pseudomanifold.

If this is right

  • Every admissible set of characteristic data determines a unique locally standard T-pseudomanifold up to weakly equivariant homeomorphism.
  • The classification applies equally to smooth cases and to singular stratified bases.
  • Equivariant maps between such pseudomanifolds correspond to maps of their characteristic data.
  • The result recovers the known classification of quasitoric manifolds as a special case.

Where Pith is reading between the lines

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

  • The same characteristic data approach may apply to other torus actions on stratified spaces beyond the conditions stated here.
  • Concrete examples such as singular toric varieties could be checked directly to test the scope of the classification.
  • Removing or weakening the base conditions might yield a still broader classification theorem.

Load-bearing premise

The base topological stratified pseudomanifolds satisfy certain unspecified conditions that allow the classification to hold.

What would settle it

Two locally standard T-pseudomanifolds with identical characteristic data but no weakly equivariant homeomorphism between them, or a pseudomanifold satisfying the conditions that cannot be recovered from any characteristic data.

Figures

Figures reproduced from arXiv: 2511.10153 by Shintaro Kuroki, Yuya Koike.

Figure 1
Figure 1. Figure 1: CΣ, C ′ Σ and spherical duals Proof of Proposition 14.2. By Proposition 14.4 and Theorem 12.1, the canonical model X(B, λ, 0) is a locally standard T-pseudomanifold. By (14.1), we have XΣ ∼= X(B, λ, 0). Therefore, by Proposition 3.5, XΣ is also a locally standard T-pseudomanifold. The class of complete toric varieties contains all projective toric varieties. Therefore, projec￾tive toric varieties are also … view at source ↗
Figure 2
Figure 2. Figure 2: Filtration of ˚c  T^pSn−1  Filtration on ˚c  T]pNp  . Consider the decomposition ^˚ TpSn−1 ∼= ]˚ TpSp × ]˚ TpNp. Here, ]˚ TpSp is equipped with the trivial filtration, while ]˚ TpNp is endowed with the following filtration: W˚ : ]˚ TpNp = W˚n−1 ⊃ W˚n−2 ⊃ · · · ⊃ W˚k ⊃ · · · ⊃ W˚i ⊃ ∅, (14.17) where W˚k := f −1  f ]˚ TpNp  ∩ Qk  . 52 [PITH_FULL_IMAGE:figures/full_fig_p052_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Filtration of ˚c  T]pNp  Hence, it is a topological stratified pseudomanifold. We now assume that the proposition holds for all dimensions less than n. Conditions Definition B.1-1 and Definition B.1-2 are straightforward to verify. For each p ∈ Qi \ Qi−1 (where 0 ≤ i ≤ n), we next construct a triple (Up, Lp, φp) satisfying Definition B.1-3. If i = n, then we have p ∈ B˚n, where B˚n denotes the interior o… view at source ↗
Figure 4
Figure 4. Figure 4: Thom space of the complex line bundle over the Hirzebruch surface [PITH_FULL_IMAGE:figures/full_fig_p058_4.png] view at source ↗
read the original abstract

We introduce the notion of a locally standard $T$-pseudomanifold, a class that generalizes both complete toric varieties and locally standard $T$-manifolds. The main goal of this paper is to show that locally standard $T$-pseudomanifolds over topological stratified pseudomanifolds satisfying certain conditions are completely classified, up to (weakly) equivariant homeomorphism, by their characteristic data. This result extends the classification of quasitoric manifolds by Davis-Januszkiewicz.

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 paper introduces the notion of a locally standard T-pseudomanifold, generalizing both complete toric varieties and locally standard T-manifolds. It proves that locally standard T-pseudomanifolds over topological stratified pseudomanifolds satisfying certain conditions (detailed in §2) are completely classified up to weakly equivariant homeomorphism by their characteristic data, consisting of a map from the strata to the Lie algebra of T together with the orbit space. This directly extends the Davis-Januszkiewicz classification of quasitoric manifolds.

Significance. This classification theorem extends a foundational result in toric topology to the broader setting of stratified pseudomanifolds with torus actions. The explicit construction of characteristic data from the space and the verification that it determines the object up to homeomorphism, without circularity, provides a solid generalization that could enable further study of singular spaces in equivariant geometry. The result is noteworthy for its direct generalization of the Davis-Januszkiewicz argument under the stated conditions on local standardness and T-action compatibility.

minor comments (3)
  1. §2: The conditions on topological stratified pseudomanifolds (local standardness and compatibility with the T-action) are load-bearing for the classification; a brief discussion of whether these conditions are minimal or can be relaxed would strengthen the statement.
  2. Introduction: The term 'weakly equivariant homeomorphism' is used in the main claim but would benefit from an explicit definition or forward reference to its precise meaning in the context of the T-action.
  3. The notation for the characteristic data (map to the Lie algebra of T) should be introduced with a clear symbol or equation number upon first use to aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. We are pleased that the work is viewed as a solid generalization of the Davis-Januszkiewicz classification to the setting of stratified pseudomanifolds.

Circularity Check

0 steps flagged

No significant circularity; classification is self-contained

full rationale

The paper defines locally standard T-pseudomanifolds and constructs their characteristic data (a map from strata to the Lie algebra of T together with the orbit space) explicitly from the given space and T-action. It then proves a bijection between such spaces (up to weakly equivariant homeomorphism) and the characteristic data, under the stated conditions on the base topological stratified pseudomanifolds. This directly generalizes the external Davis-Januszkiewicz classification for quasitoric manifolds without any reduction of the central claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The derivation chain consists of explicit constructions and compatibility verifications that are independent of the target classification result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The central claim rests on the newly introduced definition of locally standard T-pseudomanifold and on the existence of characteristic data that serves as a complete invariant under the stated conditions.

invented entities (1)
  • locally standard T-pseudomanifold no independent evidence
    purpose: Generalizes complete toric varieties and locally standard T-manifolds to include stratified cases
    New class defined in the paper to enable the classification theorem

pith-pipeline@v0.9.0 · 5374 in / 1158 out tokens · 31481 ms · 2026-05-17T22:20:57.201816+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On $3$-dimensional locally standard $T$-pseudomanifolds

    math.GT 2026-05 unverdicted novelty 5.0

    In dimensions ≤3, locally standard T-pseudomanifolds are classified by characteristic data without the homotopy equivalence condition, and manifold cases are characterized via orbit spaces.

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