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

The sphericity of the complex of non-degenerate subspaces