  We prove that the complex of proper non-trivial non-degenerate subspaces of a
finite-dimensional vector space endowed with a non-degenerate sesquilinear form
is homotopy equivalent to a wedge of spheres. Additionally, we show that the
same is true for a slight generalization, the so-called generalized Phan
geometries of type A_n. These generalized Phan geometries occur as relative
links of certain filtrations. Their sphericity implies finiteness properties of
suitable arithmetic groups and allows for a revision of Phan's
group-theoretical local recognition of suitable finite groups of Lie type with
simply laced diagram.
