Seminar

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

#### Robin Sroka - McMaster University

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\$.

Organizers
Gregory Arone
Stockholm University
Tilman Bauer
KTH Royal Institute of Technology
Alexander Berglund
Stockholm University
Søren Galatius
University of Copenhagen
Jesper Grodal,
University of Copenhagen
Thomas Kragh
Uppsala University

# ProgramContact

Alexander Berglund

alexb@math.su.se

Søren Galatius

galatius@math.ku.dk

Thomas Kragh

thomas.kragh@math.uu.se

# Otherinformation

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