Abstract
Let 𝐺 be a finite group and p𝑝 a prime. The Brown complex of G𝐺 at the prime p𝑝 is the simplicial complex associated with the poset of non-trivial p𝑝-subgroups of G𝐺. The topology of this complex plays a crucial role in understanding the algebraic p𝑝-local structure of the group and has significant implications for related representation theoretic questions. Quillen has shown that the Brown complex of a finite reductive group defined over a field of characteristic p𝑝 is homotopy equivalent to the Tits building. In this talk, we consider the case of a finite reductive group defined over a field of characteristic different from p𝑝 and show that the homotopy type of the Brown complex can be described in terms of the generic Sylow theory introduced by Broué and Malle.