Some Results About the Quillen Complex ofSp2n(q)

The structure of the Quillen complex ofSp2n(q) atp, denoted p(Sp2n(q)), is known whenpis the characteristic prime. In this paper it is shown that ifp (q − 1) then p(Sp2n(q)) is Cohen–Macaulay of dimension (n − 1). Furthermore, ifd ≥ 3 is the order ofqin /p anddis odd, it is shown that p(Sp2n(q)) is simply connected whenevermp(Sp2n(q)) ≥ 3. It is also shown that the order complex of proper, nondegenerate subspaces of a 2n-dimensional symplectic space over q—ordered by inclusion—is Cohen–Macaulay of dimension (n − 2) ifq ≠ 2.