The sphericity of the complex of non-degenerate subspaces

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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 Aₙ. 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.