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arxiv: 2407.11313 · v2 · submitted 2024-07-16 · 🧮 math.AT · math.AG· math.CO

Real toric manifolds associated with chordal nestohedra

Suyoung Choi , Younghan Yoon This is my paper

Pith reviewed 2026-05-23 23:14 UTC · model grok-4.3

classification 🧮 math.AT math.AGmath.CO
keywords real toric manifoldschordal nestohedraEL-shellabilityalternating permutationsBetti numbersbuilding setsa-numberchordal graphs
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The pith

The Betti numbers of real toric manifolds from chordal nestohedra equal the number of alternating B-permutations.

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

The authors prove that the poset induced by any chordal building set is EL-shellable. This fact supplies an explicit formula for the rational Betti numbers of the associated real toric manifold in terms of alternating B-permutations. The same count therefore gives the a-number of the underlying chordal graph. A reader cares because the result replaces a topological calculation with a purely combinatorial one that applies to all chordal graphs and their nestohedra.

Core claim

For a chordal building set B the order complex of the induced poset is EL-shellable. Consequently the rational Betti numbers of the corresponding real toric manifold are given by counting alternating B-permutations, and this count equals the a-number of any chordal graph.

What carries the argument

Alternating B-permutations for a chordal building set B, which arise from the EL-shellability of the induced poset and directly give the Betti numbers.

If this is right

  • The computation of Betti numbers reduces to counting alternating permutations.
  • The a-number of every chordal graph equals the number of its alternating permutations.
  • Explicit Betti numbers are obtained for real Hochschild varieties corresponding to Hochschild polytopes.
  • Any chordal nestohedron admits this combinatorial description of its real toric manifold's topology.

Where Pith is reading between the lines

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

  • Similar shellability arguments might apply to other classes of building sets beyond chordal ones.
  • The permutation count could be linked to known combinatorial invariants of chordal graphs such as perfect elimination orders.
  • These formulas might extend to integral coefficients or other topological invariants if the shellability lifts appropriately.

Load-bearing premise

The rational Betti numbers of the real toric manifold are completely determined by the homology of the order complex of the poset induced by the chordal building set.

What would settle it

A direct calculation of the rational Betti numbers for the real toric manifold of a small chordal nestohedron, such as one from a path graph, that fails to match the count of alternating B-permutations.

Figures

Figures reproduced from arXiv: 2407.11313 by Suyoung Choi, Younghan Yoon.

Figure 1
Figure 1. Figure 1: Categories of nestohedra counting problem, whereas the original definition of a(G) is described recursively. In particular, the results of Corollary 1.2 are equal to the a-numbers of complete graphs, path graphs, and star graphs, which correspond permutohedra, associahedra, and stellohedra, respectively. Corollary 1.2. (1) For a complete graph Kn+1 having n + 1 vertices, βk(XR B(Kn+1) ) = X I∈( [n+1] 2k ) … view at source ↗
Figure 2
Figure 2. Figure 2: Chordal nestohedron PB One can see that both (KB)ω1+ω2 and (KB){1,2} are the full subcomplex induced by the set {J ∈ B: |J ∩ {1, 2}| ≡ 1 (mod 2)} = {{1}, {2}, {1, 4}, {1, 3, 4}, {2, 3, 4}}. In this case, (KB)Iω is contractible, and, hence, β˜ ∗((KB)Iω ) vanishes. Therefore, it contribute nothing to the Betti number of XR B . If we take ω = ω1 + ω2 + ω3, the corresponding even subset is Iω = {1, 2, 3, 4}. O… view at source ↗
Figure 3
Figure 3. Figure 3: A maximal chain of 2-lighted 4-shades 1 2 1 3 23 12 3 2 1 3 2 13 13 2 1 3 2 2 1 3 1 2 3 1 3 2 3 12 23 1 123 1 2 1 1 21 12 1 2 1 1 2 11 11 2 1 1 2 2 1 1 1 2 1 1 1 2 1 12 21 1 121 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 11 1 2 1 2 111 12 1 1 1 1 11 11 1 2 1 21 3 1 2 111 12 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Hoch(m, n) for (m, n) = (3, 0),(2, 1),(1, 2) and (0, 3) Let Bm,n be the set consisting of non-empty subsets I of [m + n] such that (2.3) |I| ≥ 2 ⇒ I ∩ [m + 1, m + n] is either ∅ or [m + r, m + n] for 1 ≤ r ≤ n, where [i, j] denotes the set of all integers ℓ such that i ≤ ℓ ≤ j. It is clear that Bm,n is a connected chordal building set on [m + n]. Proposition 2.5. For non-negative integers m and n, the nest… view at source ↗
Figure 5
Figure 5. Figure 5: E(PbB2,4 ) and µB2,4 Proof. Take any interval [I, J] of PbB, where I ( J. Let C1, . . . , Cr be all connected components of the restricted building set B|J. We assume that max C1 > max C2 > · · · > max Cr for all 1 ≤ i ≤ r. Since both B|I and B|J have no connected component of odd order, each cardinality of Ci \ I, denoted by 2ki , is even. Then the cardinality of J \ I, denoted by 2N, is 2k1 +· · ·+ 2kr. … view at source ↗
Figure 6
Figure 6. Figure 6: Venustus building sets 7.1. Venustus building sets. Let us briefly summarize all the steps for the main proof. For a connected chordal building set B on [n + 1] = {1, . . . , n + 1}, we have that βk(X R B ) = X I⊂[n+1] |I| is even βe k−1((KB)I ) (by (2.2) as a variant of Theorem 2.1) = X I⊂[n+1] |I| is even βe k−1(Kodd B|I ) (required the flagness of KB, and by Lemma 5.2) = X I⊂[n+1] |I| is even βe |I|−k−2… view at source ↗
read the original abstract

This paper investigates the rational Betti numbers of real toric manifolds associated with chordal nestohedra. We consider the poset topology of a specific poset induced from a chordal building set, and show its EL-shellability. Based on this, we present an explicit description using alternating $\mathcal{B}$-permutations for a chordal building set $\mathcal{B}$, transforming the computing Betti numbers into a counting problem. This approach allows us to compute the $a$-number of a finite simple graph through permutation counting when the graph is chordal. In addition, we provide detailed computations for specific cases such as real Hochschild varieties corresponding to Hochschild polytopes.

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

1 major / 0 minor

Summary. The paper claims to compute the rational Betti numbers of real toric manifolds associated to chordal nestohedra by establishing EL-shellability of a poset induced from a chordal building set B, then deriving an explicit formula for these Betti numbers in terms of alternating B-permutations; this is said to reduce computation of the a-number of a chordal graph to a permutation-counting problem, with explicit calculations supplied for Hochschild varieties.

Significance. If the identification between manifold Betti numbers and poset homology is valid and the EL-shellability proof is complete, the work supplies a direct combinatorial method for these invariants that applies to all chordal graphs, which would be a useful addition to the literature on toric topology and nestohedra.

major comments (1)
  1. [Abstract] Abstract, paragraph 2: the central step asserting that the rational Betti numbers of the real toric manifold are completely determined by the (co)homology of the order complex of the poset induced by the chordal building set is stated without a proof, reference, or spectral-sequence argument; this identification is load-bearing for the claim that EL-shellability yields the alternating-B-permutation formula.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for identifying a point that improves the clarity of the abstract. We address the major comment below and will incorporate the suggested change in the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: the central step asserting that the rational Betti numbers of the real toric manifold are completely determined by the (co)homology of the order complex of the poset induced by the chordal building set is stated without a proof, reference, or spectral-sequence argument; this identification is load-bearing for the claim that EL-shellability yields the alternating-B-permutation formula.

    Authors: We agree that the abstract would be strengthened by an explicit reference to the source of this identification. The connection between the rational Betti numbers of the real toric manifold and the (co)homology of the order complex of the poset induced by the building set is a standard result in toric topology, following from the spectral sequence relating the cohomology of real moment-angle complexes to the homology of the associated poset (as established in the literature on nestohedra and real toric manifolds, e.g., via the work of Buchstaber-Panov and subsequent papers on building sets). This is recalled with citations in the introduction and Section 2 of the manuscript. To address the referee's concern directly in the abstract, we will revise paragraph 2 to include a brief parenthetical reference to the relevant theorem establishing the identification. This revision makes the logical dependence explicit without altering the paper's main results. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained combinatorial argument

full rationale

The paper establishes EL-shellability of the poset induced by a chordal building set directly from its definition, then converts the resulting (co)homology computation into an explicit count of alternating B-permutations. No equation or claim reduces the Betti numbers back to a fitted parameter, a self-defined quantity, or a load-bearing self-citation whose content is itself unverified; the identification between manifold Betti numbers and order-complex homology is treated as an input premise rather than derived within the paper by circular construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a pure combinatorial proof relying on standard definitions from poset topology and building-set theory. No free parameters are introduced. The only axioms are background results in algebraic combinatorics.

axioms (2)
  • standard math EL-shellability of a poset implies that its order complex is homotopy equivalent to a wedge of spheres whose number is given by the number of decreasing chains or equivalent combinatorial counts.
    Invoked when the authors conclude that Betti numbers reduce to a counting problem after proving EL-shellability.
  • domain assumption The rational Betti numbers of the real toric manifold associated to a building set equal the Betti numbers of the order complex of the induced poset.
    This correspondence is presupposed when the permutation count is said to compute the manifold Betti numbers.

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