The fundamental group of the Quillen complex of the symmetric group

Abstract

In this paper, we investigate the fundamental group of the Quillen complex of the symmetric group Δ A p ( S n ) . We show that when p is an odd prime the simplicial complex Δ A p ( S n ) is simply connected if and only if 3 p + 2 ⩽ n < p 2 or n ⩾ p 2 + p . Furthermore, we determine the fundamental group π 1 ( A p ( S n ) ) in all cases except those where p ⩾ 5 and n ∈ { 3 p , 3 p + 1 } and that where p = 3 and n = 10 .