Quotients of the order complex by subgroups of the Young subgroup

In this paper we prove that the space Δ ( Π ¯ n ) / G is homotopy equivalent to a wedge of spheres of dimension n − 3 for all natural numbers n ⩾ 3 and all subgroups G ⊂ S 1 × S n − 1 . Our argument makes use of Discrete Morse Theory. Furthermore we find a simple formula to compute the number of spheres. For example Δ ( Π ¯ n ) / S 1 × S n − 1 is homotopy equivalent to a sphere.