Robin Sroka: Postdoc day, Seminar 4: On the high-dimensional rational cohomology of special linear groups

Abstract: Work of Borel–Serre implies that the rational cohomology of \(\operatorname{SL}_n(\mathbb{Z})\) satisfies a duality property, which is analogous to Poincaré duality for manifolds. In particular, the rational cohomology of \(\operatorname{SL}_n(\mathbb{Z})\) vanishes in all degrees above its virtual cohomological dimension \(v_n = {n \choose 2}\). Surprisingly, the highest two possibly non-trivial rational cohomology groups also vanish, if \(n \geq 3\). In the top-degree \(v_n\) this is a result of Lee–Szczarba and in codimension one \(v_n - 1\) a theorem of Church–Putman. In this talk, I will discuss work in progress with Brück–Miller–Patzt–Wilson on the rational cohomology of \(\operatorname{SL}_n(\mathbb{Z})\) in codimension two \(v_n - 2\).