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arxiv: 2606.15241 · v2 · pith:B2COBDJMnew · submitted 2026-06-13 · 🧮 math.CO · math.AT· math.MG

Filtered order complexes and magnitude homology of finite graded posets

Yoh Kitajima This is my paper

Pith reviewed 2026-06-27 04:31 UTC · model grok-4.3

classification 🧮 math.CO math.ATmath.MG
keywords order complexmagnitude homologygraded posetfiltered subcomplexshellable posetgeometric semilatticesimplicial subdivisionclosed manifold
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The pith

Rank-filtered subcomplexes of order complexes in finite graded posets match the homology of an underlying closed manifold except in the top dimension, where the group is a nontrivial free abelian group.

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

The paper defines filtered subcomplexes of the order complex of a finite graded poset by cutting along the rank function and studies their topology. When the full order complex is a simplicial subdivision of a closed manifold, these filtered pieces have the same homology as the manifold in every dimension except the highest, where a free abelian group appears instead. The same filtered pieces stay shellable whenever the original poset is shellable, and for geometric semilattices they are homotopy equivalent to a wedge of spheres all of one dimension. These facts are presented as properties that also inform the magnitude homology of the poset.

Core claim

For posets whose order complexes are simplicial subdivisions of closed manifolds, the homology groups of these subcomplexes agree with that of the underlying manifold except for the top dimension, where it is a nontrivial free abelian group. For shellable graded posets each subcomplex is shellable; for geometric semilattices each is homotopy equivalent to a nontrivial wedge sum of spheres of the same dimension.

What carries the argument

The rank-filtered subcomplexes obtained by partitioning the order complex according to the grading of the poset.

If this is right

  • Homology calculations for the filtered layers reduce to the known homology of the manifold plus one extra free abelian summand in top degree.
  • Shellability of the original graded poset is inherited by every filtered subcomplex.
  • Geometric semilattices produce filtered pieces that are all homotopy equivalent to wedges of spheres of identical dimension.
  • Magnitude homology of the poset receives topological information from the filtered order complexes.

Where Pith is reading between the lines

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

  • The filtration may give a practical way to compute magnitude homology by reducing it to manifold homology plus a single correction term.
  • The same filtered pieces could be used to study other invariants that are sensitive to shellability or homotopy type in combinatorial settings.
  • If the grading is relaxed or the poset is allowed to be infinite, the statements would require new proofs or counter-examples.

Load-bearing premise

The poset must be finite and graded so that the rank function cleanly partitions the order complex into layers whose topology can be compared with the manifold or shellability properties.

What would settle it

Compute the homology of one filtered subcomplex for a concrete finite graded poset whose order complex subdivides a sphere or other closed manifold and check whether any non-top-dimensional group differs from the manifold's homology.

Figures

Figures reproduced from arXiv: 2606.15241 by Yoh Kitajima.

Figure 1
Figure 1. Figure 1: The magnitude homology of P and the homotopy types of ∆(k) (P) (0 ≤ k ≤ 2) are as follows: MH2 i (P) ∼= 0 (i = 0, 1, 2), MH1 i (P) ∼= ( Z 8 (i = 1) 0 (otherwise), MH0 0 (P) ∼= Z 8 , (2.20) ∆(2)(P) ≃ S 1 , ∆(1)(P) ≃ S 1 , ∆(0)(P) ≃ _ 7 S 0 . (2.21) The equality of (2.18) holds in the case k = 2. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Hasse diagram of P. 1 2 3 4 5 6 7 8 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: face poset of K 4 Magnitude homology and filtered order complexes of shellable graded posets. 4.1 Shellablity of simplicial complexes and posets A simplicial complex ∆ is pure or pure dimensional of n if all its maximal simplices have the same dimension n. Definition 4.1 ([12], Definition 12.1). A simplicial complex ∆ is called shellable if its max￾imal simplices can be arranged in linear order F1, . . . ,… view at source ↗
read the original abstract

In this paper, we study the family of subcomplexes of the order complexes of finite graded posets, defined via its rank function. We address three main topics. (1) We describe the general topological properties of these subcomplexes in relation to magnitude homology of graded posets. (2) For posets whose order complexes are simplicial subdivisions of closed manifolds, we show that the homology groups of these subcomplexes agree with that of the undelying manifold except for the top dimension, where it is a nontrivial free abelian group. (3) For shellable graded posets, we prove that each of the subcomplexes are also shellable. Moreover, in the case of geometric semilattices, we show that each subcomplexes are homotopy equivalent to a nontrivial wedge sums of spheres of the same dimension.

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 studies the family of rank-filtered subcomplexes of the order complexes of finite graded posets. It relates these to magnitude homology, proves that when the order complex is a simplicial subdivision of a closed manifold the homology of the subcomplexes agrees with that of the manifold except in top dimension (where it is a nontrivial free abelian group), shows that shellability of the poset implies shellability of each filtered subcomplex, and shows that for geometric semilattices each subcomplex is homotopy equivalent to a nontrivial wedge of spheres of fixed dimension.

Significance. If the claims hold, the work supplies a natural filtration on order complexes that interacts cleanly with magnitude homology and preserves or controls standard topological invariants (homology, shellability, homotopy type) under the stated hypotheses. The results appear to rest on classical poset-topology techniques applied to the rank filtration, which is a standard construction for graded posets; the manifold and geometric-semilattice cases extend known subdivision and shellability results in a uniform way.

minor comments (3)
  1. [Abstract] Abstract, line 3: 'undelying' is a typographical error and should read 'underlying'.
  2. [Abstract] Abstract, sentence on shellable graded posets: 'each of the subcomplexes are also shellable' contains a subject-verb agreement error; rephrase for grammatical correctness.
  3. [Abstract] Abstract, final sentence: 'each subcomplexes are homotopy equivalent to a nontrivial wedge sums of spheres' contains multiple grammatical issues ('each subcomplex is', 'wedge sum', 'of spheres of the same dimension').

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on rank-filtered subcomplexes, their relation to magnitude homology, and the shellability and homotopy-type statements. We appreciate the recommendation for minor revision and will incorporate any minor editorial changes in the revised version.

Circularity Check

0 steps flagged

No circularity; claims rest on external topological hypotheses

full rationale

The paper defines filtered subcomplexes via the rank function on finite graded posets and states three classes of results: general topological properties relating to magnitude homology, homology agreement with underlying manifolds (except top dimension) when the order complex subdivides a closed manifold, and preservation of shellability (plus homotopy equivalence to wedges of spheres for geometric semilattices). These are presented as theorems whose hypotheses are external (poset finiteness, grading, manifold subdivision property, shellability). No equations appear that equate a derived quantity to a fitted input by construction, and no load-bearing self-citation chain is invoked to justify uniqueness or an ansatz. The derivation chain therefore remains self-contained against standard poset-topology benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, invented entities, or non-standard axioms; all background appears to be standard graded-poset and simplicial-complex theory.

pith-pipeline@v0.9.1-grok · 5666 in / 1250 out tokens · 36395 ms · 2026-06-27T04:31:23.628449+00:00 · methodology

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Reference graph

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