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

2019/10/25 by Mathieu Dutour Sikirić, Sikirić, Mathieu Dutour, Philippe Elbaz–Vincent +5 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1910.11598

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 K₈(Z) = 0. We deduce from it new knowledge on the Kummer-Vandiver conjecture.

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