VORONOI COMPLEXES IN HIGHER DIMENSIONS, COHOMOLOGY OF $GL_N (Z)$ FOR $N\ge 8$ AND THE TRIVIALITY OF $K_8 (Z)$
Résumé
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 $K_8 (Z) = 0$, providing new knowledge on the Kummer-Vandiver conjecture.
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