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

Date: 2022-01-21

Time: 15:50 - 16:20


Robin Sroka


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 $v_n – 2$.