RÉSUMÉ

Voronoi has attached to the space $C_N$ of positive definite $N$-dimensional quadratic forms over $\Bbb R$ a $G$-equivariant cell deomposition into so called perfect forms, where $G=GL_N(\Bbb Z)$, which he showed to have only finitely many $G$-representatives. He extended $C_N$ by adding certain boundary components to it, and after taking the quotient by the action of $G$ one obtains a finite dimensional $G$-equivariant cell complex whose homology computes the homology of $G$ (with coefficients in the Steinberg module). For $N\leq 7$, there is a complete description of the underlying perfect forms known, and thus, by successive intersection of the corresponding cells and computation of the differential, the cell complex can in principle be made explicit. Our program, based on the software package GP-PARI, has built the cell complex so far for $N\leq 6$. We deduce the homology of this complex and obtain from this (up to small primes) the cohomology of $GL_N(\Bbb Z)$ with trivial coefficients, for $N\leq 6$. Furthermore, we use the information exhibited in the process to obtain results on the structure of $K_N(\Bbb Z)$ for $N=5$ and $N=6$.

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