Homotopy commutativity in quasitoric manifolds

We prove that the loop space of a quasitoric manifold is homotopy commutative if and only if the underlying polytope is (Δ³)ⁿ and the characteristic matrix is equivalent to a matrix of certain type. We also construct for each n≥ 2 and a positive integer k, a quasitoric manifold $M(k,n)$ over (Δ³)ⁿ such that its loop space is homotopy commutative if and only if k is even, where every quasitoric manifold over Δ³ is equivalent to C P³ whose loop space is homotopy commutative.