## $K_8(\mathbb{Z})=0$

A proof of the result announced by Elbaz-Vincent and mentioned above can now be found in:

- Mathieu Dutour-Sikirić, Philippe Elbaz-Vincent, Alexander Kupers, and Jacques Martinet,
*Voronoi complexes in higher dimensions, cohomology of $GL_N(\mathbb{Z})$ for $N\geq8$ and the triviality of $K_8(\mathbb{Z})$*, arXiv:1910.11598v1 (submitted on October 25, 2019)

A shorter proof, which (unlike the one above) does not require additional computer calculations, is presented in:

The very last remark in Kupers’s short note addresses the case of $K_{12}(\mathbb{Z})$:

*Remark 2.6.* If Conjecture 2 of [CFP14] were true with coefficients in $\mathbb{Z}[1/((n+1)!)]$ instead of $\mathbb{Q}$, it could be used to prove that $K_{12}(\mathbb{Z})=0$ in a similar manner.

[CFP14] Thomas Church, Benson Farb, and Andrew Putman, *A stability conjecture for the unstable cohomology of $SL_n(\mathbb{Z})$, mapping class groups, and $Aut(F_n)$*, Algebraic topology: applications and new directions, 55–70, Contemp. Math., vol. 620, Amer. Math. Soc., 2014, MR3290086