Explicit canonical cycle at the virtual cohomological dimension of $\mathrm{SL}_n(\mathbb{Z})$ through Voronoi complex

Alejandro de la Torre Dur\'an

arxiv: 2604.03743 · v2 · submitted 2026-04-04 · 🧮 math.GT

Explicit canonical cycle at the virtual cohomological dimension of SL_n(mathbb{Z}) through Voronoi complex

Alejandro de la Torre Dur\'an This is my paper

Pith reviewed 2026-05-13 17:02 UTC · model grok-4.3

classification 🧮 math.GT

keywords Voronoi complexSL_n(Z)virtual cohomological dimensioncanonical cyclearithmetic groupsrational cohomologypolyhedral tessellationshomology generator

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The pith

An explicit canonical cycle is constructed in the top-dimensional homology of the Voronoi complex for SL_n(Z).

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

The paper constructs an explicit canonical cycle in the top-dimensional homology of the Voronoi complex associated with SL_n(Z). This cycle is shown to generate the rational cohomology of SL_n(Z) at its virtual cohomological dimension. A sympathetic reader cares because earlier identifications relied on computation and conjecture, while this provides a direct geometric proof. The method uses rigidity of Voronoi tessellations and introduces an abstract framework for polyhedral tessellations under group actions on convex cones.

Core claim

We construct an explicit canonical cycle in the top-dimensional homology of the Voronoi complex associated with an arithmetic group. This cycle relates to the cohomology of SL_n(Z) with rational coefficients at the virtual cohomological dimension. This cycle has been previously identified in computational works and conjectured to provide an intrinsic generator. Our approach relies on a geometric rigidity property of Voronoi tessellations. Furthermore, an abstract framework for polyhedral tessellations of convex cones under group actions is established, elucidating the underlying mechanism of the construction of such cycles.

What carries the argument

Voronoi complex of positive definite quadratic forms under SL_n(Z) action, with its top homology generated by a canonical cycle via geometric rigidity.

Load-bearing premise

Voronoi tessellations exhibit geometric rigidity under the SL_n(Z) action, so that top-dimensional cells form orbits that yield a well-defined cycle without ambiguity.

What would settle it

For a small value such as n=4, a direct homology computation in the Voronoi complex showing the constructed cycle is homologous to zero or does not span the top homology group over the rationals.

read the original abstract

We construct an explicit canonical cycle in the top-dimensional homology of the Voronoi complex associated with an arithmetic group. This cycle relates to the cohomology of SL$_n(\mathbb{Z})$ with rational coefficients at the virtual cohomological dimension. This cycle has been previously identified in computational works and conjectured to provide an intrinsic generator. Our approach relies on a geometric rigidity property of Voronoi tessellations. Furthermore, an abstract framework for polyhedral tessellations of convex cones under group actions is established, elucidating the underlying mechanism of the construction of such cycles.

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.

Desk Editor's Note private letter to a colleague

The paper turns the computational conjecture into an explicit geometric cycle using Voronoi rigidity, but the invariance step under the full group action still needs a direct check.

read the letter

The main advance is an explicit construction of a canonical cycle in the top-dimensional homology of the Voronoi complex for SL_n(Z). It takes the cycle that earlier computations had identified and conjectured to be intrinsic and defines it directly from the geometry of the tessellation via a rigidity property under the group action. This supplies a concrete generator for the rational cohomology at the virtual cohomological dimension, which is the part that matters for people who want to compute or use these classes explicitly. They also set up an abstract framework for how group actions produce polyhedral tessellations of convex cones, and that part looks reusable for other arithmetic groups. The framework clarifies the mechanism without relying on case-by-case fitting, which is a clean step forward. The soft spot is the verification that the constructed cycle remains invariant and generates the homology under the complete SL_n(Z) action. The abstract framework is in place, but the specific top-dimensional case appears to lean on the rigidity holding exactly as needed; a direct invariance argument spelled out for this setting would make the canonicity claim stand more independently of the prior computational data. Minor gaps like that are common in geometric constructions and do not sink the overall approach. This paper is for specialists working on the cohomology of arithmetic groups, especially those using Voronoi or reduction-theoretic methods. Anyone trying to move from computational observations to explicit geometric generators will find it useful. It deserves peer review because the explicit construction and the general framework address a standing conjecture with a new angle, even if the invariance details could be tightened.

Referee Report

1 major / 1 minor

Summary. The manuscript constructs an explicit canonical cycle in the top-dimensional homology of the Voronoi complex associated to the arithmetic group SL_n(Z). This cycle is asserted to generate the rational cohomology of SL_n(Z) at its virtual cohomological dimension. The construction proceeds from a geometric rigidity property of Voronoi tessellations under the group action and is supported by an abstract framework for polyhedral tessellations of convex cones.

Significance. If the central construction is verified, the result supplies an explicit, geometrically defined generator for the top homology group, confirming conjectures arising from earlier computational work and furnishing a geometric mechanism for the cohomology of SL_n(Z) at vcd. This would strengthen the link between Voronoi complexes and the stable cohomology of arithmetic groups.

major comments (1)

[Abstract and construction section] The geometric rigidity property of Voronoi tessellations (invoked in the abstract and the main construction) is used to guarantee that the proposed top-dimensional cycle is invariant and canonical under the full SL_n(Z) action. However, the manuscript supplies no direct, self-contained verification of this invariance for the specific cycle at vcd; the argument appeals to the abstract framework without an explicit check that the cycle generates the homology independently of prior computational data. This step is load-bearing for the canonicity claim.

minor comments (1)

[Framework section] The abstract framework for polyhedral tessellations of convex cones is introduced but its precise relation to the SL_n(Z) case could be clarified with an explicit statement of which axioms are used and which are verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the importance of a self-contained verification of the invariance property. We address the major comment below and will incorporate the suggested clarification in the revised version.

read point-by-point responses

Referee: [Abstract and construction section] The geometric rigidity property of Voronoi tessellations (invoked in the abstract and the main construction) is used to guarantee that the proposed top-dimensional cycle is invariant and canonical under the full SL_n(Z) action. However, the manuscript supplies no direct, self-contained verification of this invariance for the specific cycle at vcd; the argument appeals to the abstract framework without an explicit check that the cycle generates the homology independently of prior computational data. This step is load-bearing for the canonicity claim.

Authors: The abstract framework for polyhedral tessellations of convex cones is developed precisely to supply a self-contained, group-action-independent argument that any cycle constructed via the geometric rigidity property is invariant under the full arithmetic group action and generates the top homology rationally. In the current manuscript this is applied to the Voronoi complex at vcd, but we acknowledge that the application step is not spelled out in sufficient detail. In the revision we will insert a new subsection (immediately following the statement of the main theorem) that carries out the verification explicitly: we list the generators of the top-dimensional cells, apply the rigidity axiom to show that the alternating sum is fixed by every element of SL_n(Z), and confirm that the resulting class is nonzero in homology by a direct computation with the cone structure, without reference to prior numerical data. This will render the canonicity claim fully self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity; construction rests on independent geometric rigidity

full rationale

The derivation constructs the explicit canonical cycle by invoking a geometric rigidity property of Voronoi tessellations under the SL_n(Z) action together with an abstract framework for polyhedral tessellations of convex cones. These are presented as external inputs rather than quantities defined in terms of the cycle or fitted to it. No equations, self-citations, or ansatzes are shown to reduce the top-dimensional homology generator back to its own inputs by construction. The approach is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract mentions no free parameters, no invented entities, and relies on a geometric rigidity property whose precise statement is not given here.

pith-pipeline@v0.9.0 · 5389 in / 1040 out tokens · 27632 ms · 2026-05-13T17:02:24.260196+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We construct an explicit canonical cycle in the top-dimensional homology of the Voronoi complex... relies on a geometric rigidity property of Voronoi tessellations... abstract framework for polyhedral tessellations of convex cones under group actions
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 0.2... (2)=0 iff S=ΣT_m and λ_σ=λ/|Γ_σ|... cancellation... self-intersecting facets... connectedness of the Voronoi graph

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Gunnells, and Mark McConnell

Avner Ash, Paul E. Gunnells, and Mark McConnell. Explicit sharbly cycles at the virtual cohomological dimension for SLn(Z).Journal of Homotopy and Related Structures, 20(3):391– 416, 2025

work page 2025
[2]

Borel and J.-P

A. Borel and J.-P. Serre. Corners and arithmetic groups.Commentarii Mathematici Helvetici, 48:436–483, 1973

work page 1973
[3]

Computing in arithmetic groups with Voronoï’s algorithm.Journal of Algebra, 435:263–285, 2015

Oliver Braun, Renaud Coulangeon, Gabriele Nebe, and Sebastian Schönnenbeck. Computing in arithmetic groups with Voronoï’s algorithm.Journal of Algebra, 435:263–285, 2015

work page 2015
[4]

V oronoi complexes in higher dimensions, cohomology of GLN(Z) for n⩾ 8 and the triviality of K8(Z).Journal of the Institute of Mathematics of Jussieu, 25(1):565–590, 2026

Mathieu Dutour Sikiri ´c, Philippe Elbaz-Vincent, Alexander Kupers, and Jacques Martinet. V oronoi complexes in higher dimensions, cohomology of GLN(Z) for n⩾ 8 and the triviality of K8(Z).Journal of the Institute of Mathematics of Jussieu, 25(1):565–590, 2026

work page 2026
[5]

Gunnells, Jonathan Hanke, Achill Schürmann, and Dan Yasaki

Mathieu Dutour Sikiri´c, Herbert Gangl, Paul E. Gunnells, Jonathan Hanke, Achill Schürmann, and Dan Yasaki. On the cohomology of linear groups over imaginary quadratic fields.Journal of Pure and Applied Algebra, 220(7):2564–2589, 2016

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[6]

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P Elbaz-Vincent, H Gangl, and C Soule. Some computations of the homology of GLn(Z) and the K-theory ofZ.Comptes Rendus Mathematique, 335(4):321–324, 2002

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[7]

Perfect forms, K-theory and the cohomology of modular groups.Advances in Mathematics, 245:587–624, 2013

Philippe Elbaz-Vincent, Herbert Gangl, and Christophe Soulé. Perfect forms, K-theory and the cohomology of modular groups.Advances in Mathematics, 245:587–624, 2013

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Springer Monographs in Mathematics

Günter Harder.Cohomology of Arithmetic Groups. Springer Monographs in Mathematics. Springer, Cham, 2026

work page 2026
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Énumération complète des classes de formes parfaites en dimen- sion 7.Annales de l’Institut Fourier, 43(1):21–55, 1993

David-Olivier Jaquet-Chiffelle. Énumération complète des classes de formes parfaites en dimen- sion 7.Annales de l’Institut Fourier, 43(1):21–55, 1993

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Jacques Martinet.Perfect Lattices in Euclidean Spaces, volume 327. 01 2003

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A generalization of V oronoï’s Theorem to algebraic lattices

Kenji Okuda and Syouji Yano. A generalization of V oronoï’s Theorem to algebraic lattices. Journal de théorie des nombres de Bordeaux, 22(3):727–740, 2010

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C. Soulé. On the 3-torsion in K 4(Z).Topology, 39(2):259–265, 2000

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Cohomology of arithmetic groups- fields medal lecture.Proceedings of the International Congress of Mathematicians (ICM 2018), pages 267–300

Akshay Venkatesh. Cohomology of arithmetic groups- fields medal lecture.Proceedings of the International Congress of Mathematicians (ICM 2018), pages 267–300

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Cohomology of arithmetic groups and periods of automorphic forms.Japan- ese Journal of Mathematics, 12(1):1–32, 2017

Akshay Venkatesh. Cohomology of arithmetic groups and periods of automorphic forms.Japan- ese Journal of Mathematics, 12(1):1–32, 2017

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Nouvelles applications des paramètres continus à la théorie des formes quadratiques

Georges V oronoi. Nouvelles applications des paramètres continus à la théorie des formes quadratiques. deuxième mémoire. recherches sur les parallélloèdres primitifs.Journal für die reine und angewandte Mathematik, 134:198–287, 1908. Univ. GrenobleAlpes, CNRS, InstitutFourier, F-38000 Grenoble, France Email address:alejandro.de-la-torre-duran@univ-grenobl...

work page 1908

[1] [1]

Gunnells, and Mark McConnell

Avner Ash, Paul E. Gunnells, and Mark McConnell. Explicit sharbly cycles at the virtual cohomological dimension for SLn(Z).Journal of Homotopy and Related Structures, 20(3):391– 416, 2025

work page 2025

[2] [2]

Borel and J.-P

A. Borel and J.-P. Serre. Corners and arithmetic groups.Commentarii Mathematici Helvetici, 48:436–483, 1973

work page 1973

[3] [3]

Computing in arithmetic groups with Voronoï’s algorithm.Journal of Algebra, 435:263–285, 2015

Oliver Braun, Renaud Coulangeon, Gabriele Nebe, and Sebastian Schönnenbeck. Computing in arithmetic groups with Voronoï’s algorithm.Journal of Algebra, 435:263–285, 2015

work page 2015

[4] [4]

V oronoi complexes in higher dimensions, cohomology of GLN(Z) for n⩾ 8 and the triviality of K8(Z).Journal of the Institute of Mathematics of Jussieu, 25(1):565–590, 2026

Mathieu Dutour Sikiri ´c, Philippe Elbaz-Vincent, Alexander Kupers, and Jacques Martinet. V oronoi complexes in higher dimensions, cohomology of GLN(Z) for n⩾ 8 and the triviality of K8(Z).Journal of the Institute of Mathematics of Jussieu, 25(1):565–590, 2026

work page 2026

[5] [5]

Gunnells, Jonathan Hanke, Achill Schürmann, and Dan Yasaki

Mathieu Dutour Sikiri´c, Herbert Gangl, Paul E. Gunnells, Jonathan Hanke, Achill Schürmann, and Dan Yasaki. On the cohomology of linear groups over imaginary quadratic fields.Journal of Pure and Applied Algebra, 220(7):2564–2589, 2016

work page 2016

[6] [6]

Some computations of the homology of GLn(Z) and the K-theory ofZ.Comptes Rendus Mathematique, 335(4):321–324, 2002

P Elbaz-Vincent, H Gangl, and C Soule. Some computations of the homology of GLn(Z) and the K-theory ofZ.Comptes Rendus Mathematique, 335(4):321–324, 2002

work page 2002

[7] [7]

Perfect forms, K-theory and the cohomology of modular groups.Advances in Mathematics, 245:587–624, 2013

Philippe Elbaz-Vincent, Herbert Gangl, and Christophe Soulé. Perfect forms, K-theory and the cohomology of modular groups.Advances in Mathematics, 245:587–624, 2013

work page 2013

[8] [8]

Springer Monographs in Mathematics

Günter Harder.Cohomology of Arithmetic Groups. Springer Monographs in Mathematics. Springer, Cham, 2026

work page 2026

[9] [9]

Énumération complète des classes de formes parfaites en dimen- sion 7.Annales de l’Institut Fourier, 43(1):21–55, 1993

David-Olivier Jaquet-Chiffelle. Énumération complète des classes de formes parfaites en dimen- sion 7.Annales de l’Institut Fourier, 43(1):21–55, 1993

work page 1993

[10] [10]

Jacques Martinet.Perfect Lattices in Euclidean Spaces, volume 327. 01 2003

work page 2003

[11] [11]

A generalization of V oronoï’s Theorem to algebraic lattices

Kenji Okuda and Syouji Yano. A generalization of V oronoï’s Theorem to algebraic lattices. Journal de théorie des nombres de Bordeaux, 22(3):727–740, 2010

work page 2010

[12] [12]

C. Soulé. On the 3-torsion in K 4(Z).Topology, 39(2):259–265, 2000

work page 2000

[13] [13]

Cohomology of arithmetic groups- fields medal lecture.Proceedings of the International Congress of Mathematicians (ICM 2018), pages 267–300

Akshay Venkatesh. Cohomology of arithmetic groups- fields medal lecture.Proceedings of the International Congress of Mathematicians (ICM 2018), pages 267–300

work page 2018

[14] [14]

Cohomology of arithmetic groups and periods of automorphic forms.Japan- ese Journal of Mathematics, 12(1):1–32, 2017

Akshay Venkatesh. Cohomology of arithmetic groups and periods of automorphic forms.Japan- ese Journal of Mathematics, 12(1):1–32, 2017

work page 2017

[15] [15]

Nouvelles applications des paramètres continus à la théorie des formes quadratiques

Georges V oronoi. Nouvelles applications des paramètres continus à la théorie des formes quadratiques. deuxième mémoire. recherches sur les parallélloèdres primitifs.Journal für die reine und angewandte Mathematik, 134:198–287, 1908. Univ. GrenobleAlpes, CNRS, InstitutFourier, F-38000 Grenoble, France Email address:alejandro.de-la-torre-duran@univ-grenobl...

work page 1908