Voronoi complexes in higher dimensions, cohomology of $GL_N(Z)$ for $N\geq 8$ and the triviality of $K_8(Z)$

TL;DR

This paper enumerates cells in Voronoi complexes for certain modular groups, computes their cohomology, proves the triviality of $K_8(Z)$, and provides new insights related to the Kummer-Vandiver conjecture.

Contribution

It introduces methods to enumerate Voronoi cells in higher dimensions and uses these to compute cohomology and prove the triviality of $K_8(Z)$, advancing understanding of algebraic K-theory.

Findings

Proved that $K_8(Z) = 0$.

Enumerated low-dimensional cells in Voronoi complexes for $N=8$ to $11$.

Provided new evidence related to the Kummer-Vandiver conjecture.

Abstract

We enumerate the low dimensional cells in the Voronoi cell complexes attached to the modular groups $SL_N(Z)$ and $GL_N(Z)$ for $N=8,9,10,11$, using quotient sublattices techniques for $N=8,9$ and linear programming methods for higher dimensions. These enumerations allow us to compute some cohomology of these groups and prove that $K_8(Z) = 0$. We deduce from it new knowledge on the Kummer-Vandiver conjecture.