# '22

36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

### "High-dimensional rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$"

#### Brück, Benjamin

In joint work with Miller-Patzt-Sroka-Wilson, we show that the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes in codimension two, i.e. $H^{{n \choose 2} -2}(\operatorname{SL}_n(\mathbb{Z});\mathbb{Q}) = 0$ for all $n \geq 3$. This generalises work of Lee-Szarba and Church-Putman. In order to prove our result, we use Borel-Serre duality that relates these cohomology groups to the homology of the associated Tits building. We construct an explicit partial resolution of the Steinberg module for $\operatorname{SL}_n(\mathbb{Z})$ using building-like simplicial complexes. https://arxiv.org/abs/2204.11967

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