Homology Stability for Unitary Groups

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.

2000 Mathematics Subject Classification: Primary 19G99; Secondary 11E70, 18G30, 19B10.

Keywords and Phrases: Poset, acyclicity, unitary groups, homology stability.

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