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

Mathieu Dutour Sikirić; Philippe Elbaz-Vincent; Alexander Kupers; Jacques Martinet

paper

Voronoi complexes in higher dimensions, cohomology of $G L_{N} (Z)$ for $N \geq 8$ and the triviality of $K_{8} (Z)$

arXiv:1910.11598

Abstract

We enumerate the low dimensional cells in the Voronoi cell complexes attached to the modular groups $S L_{N} (Z)$ and $G L_{N} (Z)$ 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.

Voronoi complexes in higher dimensions, cohomology of GLN​(Z) for N≥8 and the triviality of K8​(Z)

Abstract

Voronoi complexes in higher dimensions, cohomology of $G L_{N} (Z)$ for $N \geq 8$ and the triviality of $K_{8} (Z)$