In this paper the homology stability for unitary groups over a ring with finite unitary stable rank is established. First we develop a 'nerve theorem' on the homotopy type of a poset in terms of a cover by subposets, where the cover is itself indexed by a poset. We use the nerve theorem to show that a poset of sequences of isotropic vectors is highly connected, as conjectured by Charney in the eighties. Homology stability of symplectic groups and orthogonal groups appear as a special case of our results.
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B. Mirzaii, W. van der Kallen, Homology stability for unitary groups. Doc. Math. 7 (2002), pp. 143–166DOI 10.4171/DM/121