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Journal of the London Mathematical Society Advance Access published online on April 15, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn088
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© 2009 London Mathematical Society

The sphericity of the complex of non-degenerate subspaces

Alice Devillers

School of Mathematics and Statistics
University of Western Australia
35 Stirling Highway
Crawley 6009
Western Australia
adevil@maths.uwa.edu.au

Ralf Gramlich

FB Mathematik
TU Darmstadt
Schlossgartenstrasse 7
64289 Darmstadt
Germany

Current address:
School of Mathematics
The University of Birmingham
Edgbaston
Birmingham
B15 2TT
United Kingdom
gramlich@mathematik.tu-darmstadt.de

Bernhard Mühlherr

Mathematisches Institut
Universität Gießen
Arndtstrasse 2
35392 Gießen
Germany
Bernhard.M.Muehlherr@math.uni-giessen.de

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.

The first author was a Collaborateur Scientifique F.R.S.-FNRS during the time when most of the work for this article was done. The second author gratefully acknowledges a Heisenberg fellowship of the Deutsche Forschungsgemeinschaft.

2000 Mathematics Subject Classification 51E24 (primary), 20G30, 20E42, 51A50 (secondary).

Received November 20, 2007; revised November 14, 2008;
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