Explicit Cycle for the Cohohomology of SL$_n(\mathbb{Z})$ through Voronoi Complex

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.

• The paper establishes that the top homology of SLₙ(Z)'s Voronoi complex is rationally one-dimensional and generated by an explicit canonical cycle.
• It employs precise geometric and combinatorial techniques to analyze boundary cancellations and group actions within the Voronoi tessellation.
• The result bridges computational observations with theoretical insights, advancing applications in algebraic K-theory and automorphic representations.

Explicit Canonical Cycle at Virtual Cohomological Dimension for $SL_n(\mathbb{Z})$ via the Voronoi Complex

Introduction and Context

This work provides a canonical and explicit generator for the cohomology of $SL_n(\mathbb{Z})$ at the virtual cohomological dimension, formulated through the top homology of the Voronoi complex. The cohomology of arithmetic groups such as $SL_n(\mathbb{Z})$ is central for diverse questions in number theory, geometry, and algebraic $K$-theory, but constructing explicit generators is only feasible in limited cases, typically when the corresponding rational cohomology is one-dimensional.

The construction builds upon Voronoi’s theory of perfect quadratic and Hermitian forms, along with the associated polyhedral tessellations of symmetric spaces. The paper confirms—by explicit, geometric, and group-theoretic methods—a conjectured formula for the top-dimensional cycle known from computational studies, and generalizes prior results valid only in specific dimensions or group cases.

Background: Perfect Forms and Voronoi Tessellations

A perfect form in this context is a positive definite quadratic (or Hermitian) form over a number field, uniquely determined (up to homothety) by its minimal vectors. Voronoi’s reduction theory partitions the cone of non-negative definite symmetric (or Hermitian) forms into a cell complex, where cells correspond to perfect forms and their faces correspond to degenerations. This makes the Voronoi decomposition a central object both for reduction theory and as a combinatorial tool in equivariant group homology computations [euclidean_lattices_martinet, Voronoi1908].

Given the group action of $SL_n(\mathbb{Z})$ (or suitable finite-index subgroups thereof) on such spaces, the associated Voronoi complex provides a means to compute the cohomology of modular groups directly via cellular methods [2002, philippe_advances].

The Voronoi Complex and Its Structural Properties

The Voronoi complex $\operatorname{Vor}_\Gamma$ is a cellular chain complex whose $k$-cells correspond to the $\Gamma$-orbits of codimension-$k$ faces of top-dimensional Voronoi cells avoiding the boundary of the cone. Orientations are tracked carefully to account for stabilizer subgroups which can act by orientation-reversing automorphisms.

The group differential is computed in terms of the incidences and orientations between faces, using combinatorial data from the tessellation and the group action, building on explicit choices of representative forms, faces, and minimal vectors. A key structural result, leveraged in the paper, is that codimension-one faces are always either shared between two distinct top cells (non-self-intersecting) or arise as the intersection of a cell with itself (self-intersecting), and these contribute differently to the boundary under the group action.

Main Theorem: Explicit Generator for the Top Homology

The central result establishes that for $\Gamma$ as a finite-index subgroup of $SL_n(\mathbb{Z})$0 (for even $SL_n(\mathbb{Z})$1 in the Euclidean case) or $SL_n(\mathbb{Z})$2 (for odd $SL_n(\mathbb{Z})$3), the top homology $SL_n(\mathbb{Z})$4 is rationally one-dimensional, generated by

$SL_n(\mathbb{Z})$5

where the sum runs over orbits of top-dimensional cells and is weighted by the inverse order of the cell's stabilizer (2604.03743). This expression is shown to always yield a canonical, nontrivial cycle. The proof relies on a careful study of local cancellation phenomena between adjacent top cells via their shared facets, as well as the global connectedness of the Voronoi graph (the 1-skeleton where vertices are top cells and edges correspond to codimension-one adjacencies).

A general abstract framework for cell decompositions of convex cones under group actions is given, in which the mechanism for cancellation of cell boundaries is formulated and proven, with Voronoi's tessellation as a primary example.

For the full linear group $SL_n(\mathbb{Z})$6 when $SL_n(\mathbb{Z})$7 is even, the absence of a canonical orientation (as some automorphisms act by orientation reversal) causes the vanishing of the analogous homology group, confirming a known phenomenon in arithmetic group cohomology.

Implications and Comparison with Prior Work

This result confirms and extends conjectures posed in previous computational and theoretical studies [philippe_advances, hermitian]. It provides an explicit intrinsic generator for the cohomology in the stable topological range for wide classes of modular groups, unifying computational observations and extending to the hermitian (imaginary quadratic fields) and higher rank cases. The formula is independent of auxiliary choices, in contrast to other combinatorial complexes such as the Sharbly complex, where canonical cycles are not uniquely determined [Sharbly].

The construction is essential for both theoretical understanding and explicit computations of group cohomology, with direct consequences for algebraic $SL_n(\mathbb{Z})$8-theory (especially in determining the groups $SL_n(\mathbb{Z})$9 for $SL_n(\mathbb{Z})$0 [K8, 2002]) and for computing automorphic periods in arithmetic geometry [Venkatesh2016, VenkateshICM2018, Harder2025]. The rigidity and functoriality of the generator open the possibility for applications in automorphic representation theory, as well as for computational advances in high-dimensional cases where enumeration of Voronoi cells remains possible [rank8, rank9].

Theoretical and Practical Significance

The explicit canonical nature of the generator ensures that equivariant top homology for $SL_n(\mathbb{Z})$1 is tractable and provides a bridge between combinatorial representation of modular symbols, group cohomology, and geometric models arising from reduction theory. The result further implies that, under Borel–Serre duality, the top homology of the Voronoi complex models the dual of the rational cohomology of arithmetic groups, thus leading to a concrete computation of $SL_n(\mathbb{Z})$2 for appropriate $SL_n(\mathbb{Z})$3 [Borel].

Beyond conceptual clarity, the explicit cycle is critical for computational implementations. Software frameworks for arithmetic groups using Voronoi’s algorithm can now explicitly represent the corresponding (co)homology generators, facilitating calculations in higher $SL_n(\mathbb{Z})$4-theory, spectral sequences, and motivic cohomology for number fields [general_setting, (2604.03743)].

Conclusion

This paper achieves the construction of a canonical, explicit, and group-theoretically intrinsic generator for the top-dimensional (virtual cohomological dimension) homology of the Voronoi complex associated to $SL_n(\mathbb{Z})$5 and comparable groups. The structural mechanism derived from Voronoi tessellations, together with rigorous group action analysis, guarantees that this generator exhausts the rational top homology in these settings.

The established framework is generalizable to a wide class of polyhedral decompositions under arithmetic group actions, suggesting avenues for further theoretical exploration as well as enhanced computational methods in arithmetic topology and $SL_n(\mathbb{Z})$6-theory.