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The sphericity of the complex of non‐degenerate subspaces

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 slightly wider class of simplicial complexes, the so‐called generalized Phan geometries of type An. These generalized Phan geometries occur as relative links of the filtration studied in Devillers and Mühlherr (Forum Math. 19 (2007) 955–970), whose sphericity implies topological finiteness properties of suitable arithmetic groups and allows for a revision of Phan's group‐theoretical local recognition (K.‐W. Phan, J. Austral. Math. Soc. Ser. A (part I) 23 (1977) 67–77; (part II), 129–146) of suitable finite groups of Lie type with simply laced diagrams.