Unstable cohomology of $\mathsf{GL}_{2n}(\mathbb{Z})$ and the odd commutative graph complex

Technical Summary: Unstable Cohomology of $GL_{2n}(\mathbb{Z})$ and the Odd Commutative Graph Complex

Problem Statement
This paper investigates the construction of explicit cohomology classes in two distinct but related mathematical domains: the compactly supported cohomology of the locally symmetric space $P_{2n}/GL_{2n}(\mathbb{Z})$ with coefficients in the orientation bundle, and the cohomology of the odd commutative graph complex $GC_3$. The authors address the scarcity of known non-trivial classes in the unstable range for $GL_{2n}(\mathbb{Z})$ (specifically for even $n$) and the limited understanding of the cohomology of $GC_3$ in degrees below $k=-3$. The central challenge is to construct explicit representatives for these classes and to establish a geometric link between the invariant differential forms on symmetric spaces and combinatorial graph invariants.

Methodology
The authors employ a unified framework based on the Pfaffian invariant form defined on the space $P_{2n}$ of positive definite symmetric matrices of rank $2n$.

1. Pfaffian Forms on Symmetric Spaces:
   The authors define a closed differential form $\phi_{2n}$ on $P_{2n}$ using the Pfaffian of the matrix $dX \cdot X^{-1} \cdot dX$, normalized by $\sqrt{\det X}$. This form transforms by the sign of the determinant under the action of $GL_{2n}(\mathbb{R})$, making it an orientation form. The authors analyze the asymptotic behavior of $\phi_{2n}$ and its products with canonical invariant forms $\beta_{4k+1} = \text{tr}((X^{-1}dX)^{4k+1})$ near the boundary of the Borel-Serre compactification. They utilize a complex of differential forms with integrable square-root singularities to extend these forms to the boundary, allowing for the definition of compactly supported cohomology classes.

2. Graph Complexes and Tropical Torelli Maps:
   The paper interprets these Pfaffian forms on the moduli space of metric graphs. For a graph $G$, the space of positive edge lengths is mapped to the space of positive definite matrices via the dual graph Laplacian $\Lambda_C$, which depends on a choice of cycle basis $C$. By pulling back the Pfaffian forms and canonical forms via this map, the authors construct differential forms on the simplex of edge lengths $\sigma_G$.

3. Integration and Cocycles:
   The authors define linear functionals $I(\omega): GC_3 \to \mathbb{R}$ by integrating the pullback of these forms over the simplex $\sigma_G$. They utilize Stokes' theorem on the compactification of the simplex to relate the coboundary of these functionals to Lie brackets in the graph complex. Specifically, they identify a "dipole sum" $m$ arising from the boundary terms, which acts as a twist in the differential.

Key Contributions and Results

• Unstable Cohomology of $GL_{2n}(\mathbb{Z})$:
  The authors construct an infinite family of non-trivial classes in the compactly supported cohomology $H^\bullet_c(P_{2n}/GL_{2n}(\mathbb{Z}); \mathcal{O})$.

  • They define a bigraded vector space $\Omega_c[\phi](2n)$ generated by "compact type" forms (products of $\beta$ forms and $\phi_{2n}$).
  • Theorem 1.1: They prove a canonical injective map from the symmetric algebra on these forms into the compactly supported cohomology.
  • Theorem 1.2: By showing these forms are primitive with respect to a Hopf algebra structure on the cohomology (recently introduced in [AMP24]), they deduce that the cohomology contains the symmetric algebra generated by these orientation forms. This provides a large amount of new unstable cohomology for $GL_{2n}(\mathbb{Z})$ where $n$ is even.

• Cohomology of the Odd Graph Complex ($GC_3$):
  The authors construct explicit cocycles in $GC_3$ using the "non-compact type" Pfaffian forms.

  • Theorem 1.3: They prove the absolute convergence of the integrals $I_G(\omega)$ for graphs with even loop numbers.
  • Theorem 1.4: They establish a Stokes' relation linking the coboundary of the cochains $I(\omega)$ to the Lie bracket with a specific dipole sum $m$, providing a geometric origin for the twisted differential on $GC_3$.
  • Theorem 1.7: Through explicit computation, they demonstrate that the class defined by the integral of the form $\beta_5$ (associated with a specific linear combination of 20 graphs with 6 loops and 12 edges) is non-zero in $H^{-6}(GC_3) \otimes \mathbb{R}$. Since $\dim H^{-6}(GC_3) = 1$, this class spans the entire cohomology group in that degree.

• Duality and Distinction:
  The paper highlights a striking duality: the "compact type" forms generate the cohomology of $GL_{2n}(\mathbb{Z})$ but vanish when integrated over graphs (due to vanishing theorems for graphs with low loop orders). Conversely, the "non-compact type" forms map to zero in the cohomology of $GL_{2n}(\mathbb{Z})$ (due to the vanishing of the Euler class) but yield non-trivial classes in $GC_3$.

Significance
The paper claims to provide the first explicit construction of non-trivial cohomology classes for $GL_{2n}(\mathbb{Z})$ in the unstable range for even $n$, explaining the structure of these classes via a symmetric algebra of Pfaffian forms. Furthermore, it offers a geometric explanation for the origin of the twisted differential in the odd graph complex and explicitly identifies a non-trivial cohomology class in $H^{-6}(GC_3)$, which was previously known only by dimension counts. The work bridges the gap between the topology of locally symmetric spaces and the combinatorics of graph complexes, revealing that the "non-compact" forms, which are trivial in the symmetric space context, are the source of non-trivial topology in the graph complex. The authors note that while the integrals for $GC_3$ can produce complex numbers involving polylogarithms, the specific non-trivial class identified in $H^{-6}(GC_3)$ simplifies to a homogeneous weight 3 expression.