Perfect forms and the cohomology of modular groups

For N=5, 6 and 7, using the classification of perfect quadratic forms, we compute the homology of the Voronoi cell complexes attached to the modular groups SL_N(\Z) and GL_N(\Z). From this we deduce the rational cohomology of those groups.

💡 Research Summary

The paper tackles the long‑standing problem of determining the rational cohomology of the arithmetic groups SLₙ(ℤ) and GLₙ(ℤ) for the ranks N = 5, 6, 7. The authors follow the Voronoi reduction theory of perfect quadratic forms: a positive‑definite quadratic form h in N variables is called perfect if its set of minimal integral vectors m(h) determines h up to a scalar. Voronoi proved that the convex hulls σ(h) of these minimal vectors give a Γ‑invariant cell decomposition of the space X*ₙ of positive quadratic forms (modulo homotheties). The cells and their intersections form a CW‑complex on which Γ = SLₙ(ℤ) or GLₙ(ℤ) acts.

Section 3 defines the “Voronoi complex” V·(Γ). For each dimension n the authors select a set Σₙ(Γ) of representatives of n‑cells whose stabilizer preserves orientation, and they generate a free abelian group Vₙ. The boundary map dₙ is given explicitly by a signed sum over oriented faces (formula (1)). By comparing with the first differential of the equivariant spectral sequence for the pair (Xₙ,∂Xₙ) they prove dₙ₋₁∘dₙ = 0, thus confirming that (V·,d) is a genuine chain complex.

A crucial homological identification is then recalled: the relative homology Hₖ(Xₙ,∂Xₙ;ℤ) vanishes except in degree N‑1, where it equals the Steinberg module Stₙ. Consequently, for any m, \