Integral cohomology of classical groups over a finite field

Abstract

Let G be one of the groups GL n , SL n , U n , SU n or Sp 2n over a finite field of characteristic p. We calculate the order of the integral Chern classes, obtained by Brauer lifting. From the cellular cochains of suitable Eilenberg–MacLane spaces, we construct a complex. Using a description of the mod l cohomology of G (l prime ≠ p), we prove thatthe homology of this complex is the integral cohomology of G, away the p-torsion. Quillen gives the mod l cohomology of GL n and Fiedorowicz and Priddy give this cohomology for the groups U n and Sp 2 n . We compute this cohomology for the special groups SL n and SU n .