Cone-decompositions of the unitary groups

The Lusternik–Schnirelmann category of a space is a homotopy invariant. Cone-decompositions are used for giving upper-bound for Lusternik–Schnirelmann categories of topological spaces. Singhof has determined the Lusternik–Schnirelmann categories of the unitary groups. In this paper I give two cone-decompositions of each unitary group for alternative proofs of Singhof's result. One cone-decomposition is easy. The other is closely related to Miller's filtration and Yokota's cellular decomposition of the unitary groups.