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REVIEW 3 major objections 4 minor 33 references

Simplicial vs. cubical spheres, polyhedral products and the Nevo-Petersen conjecture

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Pith's one-line read Flag Murai spheres are exactly the spheres whose dual simple polytope is a cube, a cube times a pentagon, a cube times a hexagon, or a cube with two adjacent edge cuts, and this classification proves the Nevo-Petersen conjecture for them.

desk verdict Flag Murai sphere classification is a real result, but Lemma 4.3 has a load-bearing gap that a referee should ask the authors to close. read the letter →

arxiv 2411.14036 v1 pith:WE32E5TS submitted 2024-11-21 math.CO math.AT

Ivan Limonchenko , Rade Živaljević This is my paper · ORCID
classification math.COmath.AT MSC 13F5555N1055S2057S12
keywords BiersphereMuraipolyhedralproductfaceringnestohedrontruncationpolytopeNevo-Petersenconjectureflagsimplicialcomplex
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper pins down the exact shapes of flag Murai spheres, the generalized Bier spheres built from a multicomplex and its Alexander dual. It proves that a Murai sphere is flag if and only if it is polytopal, with the dual simple polytope being one of four rigid types: a cube $I^n$, or a product $I^n\times P_5$, $I^n\times P_6$, or $I^n\times Q^3_2$, where $P_5$ and $P_6$ are the pentagon and hexagon and $Q^3_2$ is the cube with two adjacent edge cuts. Because every such polytope is a flag nestohedron, equivalently a 2-truncated cube, the classification feeds into a known result on gamma-vectors and settles the Nevo-Petersen conjecture for all flag Murai spheres. The same methods reprove the flag classification for classical Bier spheres and give a companion algebraic dichotomy: a Bier sphere is Golod exactly at simplex boundaries and minimally non-Golod exactly when its dual polytope is a simplex with some vertices cut off. A final construction realizes each Bier sphere as the boundary of the intersection of two polyhedral products, producing its canonical cubical model.

What carries the argument

The central object is the Murai sphere $\mathrm{Bier}_c(M)$, a simplicial sphere obtained from a proper $c$-multicomplex $M$ and its Alexander dual $M^\vee$ by a deleted-join construction. The workhorse is the external description of its Stanley-Reisner ideal: $I_{\mathrm{Bier}_c(M)} = \mathrm{pol}(I_c(M)) + \mathrm{pol}^*(I_c(M^\vee)) + \mathrm{pol}(x_1^{c_1+1},\ldots,x_m^{c_m+1})$, where $\mathrm{pol}$ and $\mathrm{pol}^*$ are the two polarizations of a monomial ideal. Flagness forces every generator of this ideal to have degree at most two, which restricts $M$ to five very simple generating sets; each is then matched, via the nerve-complex correspondence and the fact that a flag nestohedron is a 2-truncated cube, to a product polytope. For the algebraic results, the other load-bearing identity is Hochster's formula identifying the Tor-algebra of a face ring with the reduced cohomology of all full subcomplexes.

What would settle it

A concrete way to test the theorem is to enumerate small multicomplexes: for instance, take $c=(2,2)$ and $M=\langle x_1^2x_2,\,x_1x_2^2\rangle$, compute the minimal non-faces of $\mathrm{Bier}_c(M)$ from the three-term ideal identity, and check whether the sphere is flag while its dual polytope lies outside the four listed families. Any flag Murai sphere whose dual is not a 2-truncated cube would refute the classification; equivalently, a flag Murai sphere of dimension at least three that is not PL-homeomorphic to the boundary of one of those four polytopal types would refute it.

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Extended reading notes

Core claim

The central discovery is an equivalence between a purely combinatorial property and a rigid polytopal family. For any proper multicomplex $M$ with exponent vector $c$, the Murai sphere $\mathrm{Bier}_c(M)$ is flag if and only if its dual simple polytope $P_{M,c}$ is a flag nestohedron of one of four types: $I^n$ for $n\geq 1$, $I^n\times P_5$, $I^n\times P_6$ for $n\geq 0$, or $I^n\times Q^3_2$ for $n\geq 0$, where $P_5$ is the pentagon, $P_6$ is the hexagon, and $Q^3_2$ is the three-dimensional cube with two adjacent edge cuts. The flag case is reduced, through the polarization description of the Stanley-Reisner ideal of $\mathrm{Bier}_c(M)$, to five very simple generating sets for the multicomplex; once those are identified, each is matched to a product polytope via the nerve-complex correspondence. As a direct corollary, every flag Murai sphere is a 2-truncated cube, so its gamma-vector is the f-vector of a flag simplicial complex and the Nevo-Petersen conjecture holds in this class. For classical Bier spheres the same route yields the identical four-type classification, and a separate theorem characterizes Golod and minimally non-Golod Bier spheres in terms of simplex boundaries and vertex cuts. The paper also constructs the cubical Bier sphere as the boundary of the intersection of two polyhedral products, giving a canonical cubical subdivision of any Bier sphere.

Load-bearing premise

The classification rests on the quoted external theorem that the Stanley-Reisner ideal of a Murai sphere is exactly the sum of the two polarizations plus the corner terms $\mathrm{pol}(x_i^{c_i+1})$; every minimal non-face computation in the case analysis uses this identity. If that theorem had hidden conditions or were incomplete, the flag classification and its Nevo-Petersen corollary would collapse.

Editorial extensions

If this is right

  • Every flag Murai sphere is polytopal, and the dual simple polytope is a flag nestohedron, i.e. a 2-truncated cube.
  • The gamma-vector of every flag Murai sphere satisfies the Nevo-Petersen conclusion: it is the f-vector of a flag simplicial complex.
  • For classical Bier spheres the theorem gives the same four polytopal types, reproving and sharpening the known flag classification.
  • A Bier sphere is Golod if and only if it is the boundary of a simplex, and minimally non-Golod if and only if its dual polytope is a simplex with $\ell$ of its vertices cut off.
  • The cubical Bier sphere construction gives a canonical cubical subdivision of any Bier sphere, embedded as the boundary of the intersection of two polyhedral products.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the classification is an iff statement, it gives a quick negative test: a Murai sphere whose dual is not a 2-truncated cube cannot be flag, so hard cases of the Nevo-Petersen conjecture must lie outside the Murai family.
  • The parametrization by five generating sets in Lemma 4.3 suggests an enumeration route: search over multicomplexes with small $|c|$ to find flag spheres that are not Murai spheres, testing how far the conjecture propagates.
  • The cubical realization $\partial Z(K,K^\vee)\cong B(K,K^\vee)_{\mathrm{cubic}}$ may allow tools from moment-angle complexes, such as Buchstaber numbers and Massey products, to be transferred to classical Bier spheres, something the paper does not pursue.
  • A natural extension, not attempted here, would ask whether Murai spheres admit an analogous minimally-non-Golod classification for their face rings, using the multicomplex generators in place of simplicial complex generators.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies combinatorial, algebraic, and topological aspects of Bier spheres and their generalizations. It states and proves a classification of flag Murai spheres as nerve complexes of flag nestohedra of four explicit types, derives the Nevo-Petersen conjecture for all flag Murai spheres, classifies Golod and minimally non-Golod Bier spheres, and introduces a cubical Bier sphere via polyhedral products, showing that its boundary is canonically isomorphic to a certain intersection of two polyhedral products. The main new results are Theorems 1.1, 1.3, 1.5, and Propositions 8.1-8.2.

Significance. If the classification of flag Murai spheres is correct, it gives a complete and very concrete answer to a natural polytopality question and settles the Nevo-Petersen conjecture for this entire class of flag homology spheres. The cubical Bier sphere construction is original and connects Bier spheres in a clean way with the theory of polyhedral products and Alexander duality. The paper also ships explicit Stanley-Reisner ideal computations for the listed cases, which is helpful for verification. However, the proof of the core classification lemma contains a load-bearing minimality gap that needs to be repaired before the main claim can be accepted.

major comments (3)
  1. [4, proof of Lemma 4.3] The proof repeatedly infers c_i = 1 from the assertion that the polarization pol(x_i^{c_i+1}) 'must have degree 2' because Bier_c(M) is flag. This inference is valid only if pol(x_i^{c_i+1}) is a minimal generator of I_{Bier_c(M)}. The manuscript itself notes after Theorem 2.13 that the three displayed families of generators need not be in one-to-one correspondence with the minimal non-faces, citing [28, Example 3.8]. In the branch where x_m^{c_m} ∈ M, the final paragraph asserts without proof that pol(x_m^{c_m+1}) must divide an element of G(I_c(M)) ∪ G(I_c(M∨)); this is exactly a minimality claim, and no argument is supplied. Since Lemma 4.3 is the only bridge from flagness to the five listed multicomplex types, this gap is load-bearing for Theorems 4.4 and 4.5.
  2. [5, proof of Theorem 5.4(a)] The proof begins with the statement that 'a triangulated sphere is Golod if and only if it is a boundary of a simplex', with no proof or reference. This is a strong 'if and only if' criterion of the same nature as the theorem being proved, and it is not immediate from the definitions of Golodness in terms of products and Massey products in Tor-algebra. If the statement is true, a proof or a precise citation must be provided; if it is false, Theorem 5.4(a) collapses. As written, the argument rests on an unstated external criterion.
  3. [4, Theorem 4.4 case analysis] In the polytopality part of Theorem 4.4, the identification of the Murai spheres in cases (3), (4), and (6) with suspensions of a 5-gon or with cross-polytopes is asserted immediately after displaying a Stanley-Reisner ideal. These identifications are needed for the 'if' direction of the theorem and for the claim that all flag Murai spheres are polytopal. An explicit isomorphism between the simplicial complex defined by the displayed ideal and the stated nerve complex, or a reference for each identification, would make this step verifiable.
minor comments (4)
  1. [Title page] The displayed title contains a typo, 'POL YHEDRAL PRODUCTS', which should read 'POLYHEDRAL PRODUCTS'.
  2. [2, Definition 2.11] The notation |c| is used from Definition 2.11 onward but is never defined; the paper should state explicitly that |c| := c_1 + ... + c_m.
  3. [3, Lemma 3.3(b)] The claim that Bier(K) is flag if and only if K and K^vee are flag is said to follow immediately from the structure of min(Bier(K)), but since the minimal non-faces of a deleted join are not literally the union of min(K) and min(K^vee), a sentence explaining the exact set of minimal non-faces would improve the exposition.
  4. [4, Theorem 4.4 case (4)] In case (4), the expression Bier(⟨{x_i^(0), x_i^(2)}, {x_k^(0), x_i^(2)}⟩ ⊔ {∅}) appears to contain a typo: the second facet should probably be {x_k^(0), x_k^(2)} rather than {x_k^(0), x_i^(2)}.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the flag Murai classification is computed from Murai's external ideal description and an independent 2D Bier-sphere classification, and the flagged Lemma 4.3 minimal-generation gap is a correctness risk, not a circular reduction.

full rationale

The derivation chain is not circular. The classification of flag Murai spheres (Lemma 4.3, Theorem 4.4) is computed from the external ideal-theoretic description of I_Bier_c(M) quoted from Murai [28, Theorem 3.6] (Theorem 2.13), from the elementary identities of Lemma 4.2, and from finite case analysis; the Nevo-Petersen corollaries (3.6, 4.5) follow from that classification together with external theorems [10] (flag nestohedra are 2-truncated cubes) and [1, Theorem 10], so no equation in the chain is equivalent by construction to the claim it is used to establish. The reliance on [24] (Limonchenko-Sergeev, a prior paper of the first author) for the two-dimensional Bier-sphere classification and for the flag-nestohedron identifications (Theorem 3.5 proof: 'classification of 2-dimensional Bier spheres, see [24, Theorem 2.16]'; Corollary 3.6; Lemma 3.4; Theorem 5.4(b), m = 4 case) is a self-citation, but it is an independent finite classification that is externally verifiable, so it is real evidence and is scored only mildly. A separate issue is flagged because the manuscript itself asserts it: after Theorem 2.13 the paper notes that 'the union of ideal generators in the r.h.s. may not be in one-to-one correspondence with the set of minimal non-faces even in the Bier sphere case' (Section 2), yet the Lemma 4.3 case analysis repeatedly infers 'as Bierc(M ) is flag... polarizations of xci+1 i ... must have degree 2, therefore ci = 1' and finally asserts that 'polarization of xcm+1 m must divide an element in G(Ic(M )) ∪ G(Ic(M ∨))' without proving that the relevant polarizations are minimal generators. If a high-degree polarization were redundant, a flag Murai sphere outside the listed cases would not be excluded, and Corollary 4.5 as stated would be unsupported; this is a missing-support (correctness) risk, not a circularity, because the conclusion does not reduce to its inputs by construction. Sections 7-8 (cubical Bier spheres and polyhedral products) are self-contained geometric constructions with no circular dependency.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper rests on external theorems about Bier spheres, Murai spheres, face ideals, and Golodness. No new constants or postulated objects are introduced. The most fragile input is the unproved assertion about Golod triangulated spheres and the dependence on the two-dimensional Bier sphere classification from [24].

assumptions (7)
  • standard math Bier(K) is a PL sphere for any simplicial complex K not equal to a full simplex.
    Introduced in [5] with PL homeomorphism in [25]; invoked in Definition 2.7 and throughout the paper to treat Bier(K) as a sphere.
  • standard math Bier_c(M) is a simplicial sphere for any proper c-multicomplex M.
    Proved in [28, Proposition 1.10] and used in Section 4 to know that Murai spheres are objects to which the flag and polytopality questions apply.
  • standard math The Stanley-Reisner ideal of Bier_c(M) equals pol(I_c(M)) + pol*(I_c(M^vee)) + pol(x_1^{c_1+1}, ..., x_m^{c_m+1}).
    Quoted as Theorem 2.13 from [28, Theorem 3.6]; this is the computational foundation of Lemma 4.3 and Theorem 4.4.
  • domain assumption The classification of two-dimensional Bier spheres from [24, Theorem 2.16] is correct.
    This result, from a paper by the first author and Sergeev, anchors the base cases in Lemma 3.4 and Theorem 3.5. It is not reproved here.
  • standard math Flag nestohedra are 2-truncated cubes, and the gamma-vector theorem of Aisbett and Volodin [1, Theorem 10] applies to them.
    Used in Corollaries 3.6 and 4.5 to deduce the Nevo-Petersen conclusion from the polytopal classification.
  • standard math Hochster's formula and the product formula (5.1) describe the Tor-algebra of a face ring.
    Used in Theorem 5.3 and in the proof of Theorem 5.4 to detect non-Golodness via full subcomplexes that contain 4-cycles or 5-cycles.
  • ad hoc to paper A triangulated sphere is Golod if and only if it is the boundary of a simplex.
    Asserted in the proof of Theorem 5.4(a) without proof or citation. It is load-bearing for the Golod classification of Bier spheres.

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Pith. "Pith review of Simplicial vs. cubical spheres, polyhedral products and the Nevo-Petersen conjecture." pith.science (2026). https://pith.science/paper/WE32E5TS

@misc{pith2026241114036,
  author       = {Pith},
  title        = {Pith review of: Simplicial vs. cubical spheres, polyhedral products and the Nevo-Petersen conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WE32E5TS}},
  note         = {Machine review of arXiv:2411.14036}
}
abstract

We prove that a Murai sphere is flag if and only if it is a nerve complex of a flag nestohedron and classify all the polytopes arising in this way. Our classification implies that flag Murai spheres satisfy the Nevo-Petersen conjecture on $\gamma$-vectors of flag homology spheres. We continue by showing that a Bier sphere is minimally non-Golod if and only if it is a nerve complex of a truncation polytope different from a simplex and classify all the polytopes arising in this way. Finally, the notion of a cubical Bier sphere is introduced based on the polyhedral product construction, and we study combinatorial and geometrical properties of these cubical complexes.

Figures

Figures reproduced from arXiv: 2411.14036 by the authors.

Figure 1
Figure 1. A “random element” of Xm. There is a quite useful, alternative definition (description) of the polyhedral product (6.1). Given a “random element” x = (xi) ∈ Xm (or alternatively a function x : [m] → X), the corresponding “hitting set” HA(x) is the set HA(x) := {i ∈ [m] | xi ∈ A} = x −1 (A). Similarly, the associated complementary “missing set” MA(x) = HA(x) c is the set MA(x) := {i ∈ [m] | xi ∈/ A} = x −1 (A c ). Pr… view at source ↗
Figure 2
Figure 2. The front complex of a cube. Each non-empty subset I = {j1, . . . , jk} ⊂ [m] can be associated both a vertex bI = (1/k)(ej1 + . . . + ejk ) of the barycentric subdivision of the simplex ∆m−1 = conv{e1, . . . , em} and the vertex eI = ej1 + . . . + ejk of the standard cube I m ⊂ R m. The correspondence bI 7→ eI is just the radial projection from the vertex e∅ = 0 of the cube. Therefore it is extended to a piecewise … view at source ↗
Figure 3
Figure 3. [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: In light of the relation Z(K, K∨ ) = [ (A,B,C)∈B(K,K∨)+ (J⩾0) A × {0} B × (I⩽0) C and (8.3), the complex Z(K, K∨) is the union of six squares. Acknowledgements. The authors are grateful to Taras Panov, Matvey Sergeev, and Aleˇs Vavpetiˇc for various fruitful discussion…

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