Quasitoric stably normally split manifolds

TL;DR

This paper characterizes when quasitoric manifolds have stably trivial complex tangent or normal bundles, introduces a new criterion based on cohomology forms, and applies it to specific classes of manifolds.

Contribution

It provides a new criterion for the TNS property of quasitoric manifolds using cohomology forms and characterizes such manifolds in terms of their moment polytope properties.

Findings

Constructed manifolds where all complex bundles are stably sums of linear bundles.

Established a criterion for TNS property via non-semidefiniteness of cohomology forms.

Applied the criterion to show the flag property of moment polytopes in certain cases.

Abstract

A smooth stably complex manifold is called a totally tangentially/normally split manifold (TTS/TNS-manifold, for short, resp.) if the respective complex tangential/normal vector bundle is stably isomorphic to a Whitney sum of complex linear bundles, resp. In this paper we construct manifolds $M$ s.t. any complex vector bundle over $M$ is stably equivalent to a Whitney sum of complex linear bundles. A quasitoric manifold shares this property iff it is a TNS-manifold. We establish a new criterion of the TNS-property for a quasitoric manifold $M$ via non-semidefiniteness of certain higher-degree forms in the respective cohomology ring of $M$. In the family of quasitoric manifolds, this generalises 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 $3$.