Quasitoric totally normally split manifolds

A smooth stably complex manifold is called a totally tangentially/normally split manifold (TTS/TNS manifold for short) if the respective complex tangential/normal vector bundle is stably isomorphic to a Whitney sum of complex line bundles, respectively. In this paper we construct manifolds $M$ such that any complex vector bundle over $M$ is stably equivalent to a Whitney sum of complex line bundles. A quasitoric manifold shares this property if and only if it is a TNS manifold. We establish a new criterion for a quasitoric manifold $M$ to be TNS via non-semidefiniteness of certain higher degree forms in the respective cohomology ring of $M$. In the family of quasitoric manifolds, this generalizes the theorem of J. Lannes about the signature of a simply connected stably complex TNS $4$-manifold. We apply our criterion to show the flag property of the moment polytope for a nonsingular toric projective TNS manifold of complex dimension $3$.