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

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arxiv 1910.11598 v1 pith:TMHH2BZZ submitted 2019-10-25 math.KT math.NT

Mathieu Dutour Sikirić , Philippe Elbaz-Vincent , Alexander Kupers , Jacques Martinet This is my paper · ORCID
classification math.KTmath.NT
keywords cohomologycomplexesdimensionsgroupshighervoronoiallowattached
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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.

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