Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The wheel classes in the locally finite homology of $\mathrm{GL}_n(\mathbb{Z})$, canonical integrals and zeta values

Francis Brown, Oliver Schnetz

TL;DR

The paper develops a comprehensive framework to compute canonical integrals associated with wheel graphs, proving they are proportional to odd zeta values and thereby yielding explicit nonzero classes in locally finite GL_n(Z) homology and in the tropical moduli spaces of curves and abelian varieties. It introduces a general formula for the invariant forms ω^{2n-1}_{Λ_G} via a detailed study of matrices with 1-form entries, reduces to symmetric cases using determinant identities, and specializes to wheel graphs to obtain explicit expressions in terms of Feynman-type integrals and zeta-values. The authors derive infinite families of graph-homology classes Xi_{m,n} with periods ζ(m)ζ(n), relate wheel integrals to regulator-type phenomena, and connect these to wheel motives, top-weight cohomology, and tropical Torelli theory. The work also provides new determinantal identities for antisymmetrised permanents and Amitsur-Levitzki-type results, offering tools potentially applicable to broader graph complexes and motivic questions. Overall, the results illuminate deep links between graph theory, arithmetic (zeta values, regulators), algebraic K-theory, and tropical geometry through explicit canonical integrals and their homological and motivic incarnations.

Abstract

We compute the canonical integrals associated to wheel graphs, and prove that they are proportional to odd zeta values. From this we deduce that wheel classes define explicit non-zero classes in: the locally finite homology of the general linear group $\GL_n(\ZZ)$ in both odd and even ranks, the homology of the moduli spaces of tropical curves, and the moduli space of tropical abelian varieties. We deduce the existence of a doubly infinite family of auxiliary classes in the even commutative graph complex.

The wheel classes in the locally finite homology of $\mathrm{GL}_n(\mathbb{Z})$, canonical integrals and zeta values

TL;DR

The paper develops a comprehensive framework to compute canonical integrals associated with wheel graphs, proving they are proportional to odd zeta values and thereby yielding explicit nonzero classes in locally finite GL_n(Z) homology and in the tropical moduli spaces of curves and abelian varieties. It introduces a general formula for the invariant forms ω^{2n-1}_{Λ_G} via a detailed study of matrices with 1-form entries, reduces to symmetric cases using determinant identities, and specializes to wheel graphs to obtain explicit expressions in terms of Feynman-type integrals and zeta-values. The authors derive infinite families of graph-homology classes Xi_{m,n} with periods ζ(m)ζ(n), relate wheel integrals to regulator-type phenomena, and connect these to wheel motives, top-weight cohomology, and tropical Torelli theory. The work also provides new determinantal identities for antisymmetrised permanents and Amitsur-Levitzki-type results, offering tools potentially applicable to broader graph complexes and motivic questions. Overall, the results illuminate deep links between graph theory, arithmetic (zeta values, regulators), algebraic K-theory, and tropical geometry through explicit canonical integrals and their homological and motivic incarnations.

Abstract

We compute the canonical integrals associated to wheel graphs, and prove that they are proportional to odd zeta values. From this we deduce that wheel classes define explicit non-zero classes in: the locally finite homology of the general linear group in both odd and even ranks, the homology of the moduli spaces of tropical curves, and the moduli space of tropical abelian varieties. We deduce the existence of a doubly infinite family of auxiliary classes in the even commutative graph complex.
Paper Structure (72 sections, 45 theorems, 257 equations, 3 figures, 2 tables)

This paper contains 72 sections, 45 theorems, 257 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Discussion and examples of wheel canonical integrals
  3. Applications to graph homology
  4. Homology of the general linear group.
  5. Homology of the moduli spaces of tropical curves and abelian varieties
  6. Wheel motives and periods
  7. Computations of canonical integrals for other graphs
  8. Strategy of proof of Theorem \ref{['thm: introMain']} and contents
  9. Acknowledgements
  10. Further geometric applications and their proofs
  11. Graph homology
  12. Homology of the general linear group: Theorems \ref{['thm: introBMclass']} and \ref{['thm: introLAgclass']}
  13. Proof of Theorem .
  14. Proof of Theorem \ref{['thm: introLAgclass']}
  15. Proof of Theorem \ref{['thm: introBMclass']} (ii).
  16. ...and 57 more sections

Key Result

Theorem 1

For all odd $n\geq 3$,

Figures (3)

  • Figure 1: The wheel $W_3$ and the general wheel $W_n$.
  • Figure 2: Locally-finite homology of $\mathcal{P}_n/\mathrm{GL}_n({\mathbb Z})$ in the known ranges SouleSL3LeeSzczarbaElbazVincentGanglSoule. See also GL8 for some partial results for $n=8$. An entry $H^{\mathrm{lf}}_i$ in the table indicates that $H^{\mathrm{lf}}_i(\mathcal{P}_n/\mathrm{GL}_n({\mathbb Z});{\mathbb R})$ is non-zero and of dimension $1$. The classes in red are spanned by $\tau_{W_3}, \tau_{W_5}, \tau_{W_7},\ldots$; the classes indicated in blue are their images under inflation.
  • Figure 3: The closed graph with $I_G(\omega^{13})=18018\zeta(7)$.

Theorems & Definitions (115)

  • Theorem 1
  • Corollary 2
  • Remark 3
  • Theorem 4
  • Corollary 5
  • Remark 6
  • Corollary 7
  • Theorem 8
  • Theorem 9
  • Remark 10
  • ...and 105 more