Vector bundles over Davis-Januszkiewicz spaces with prescribed characteristic classes

TL;DR

This paper constructs specific vector bundles over Davis-Januszkiewicz spaces with prescribed characteristic classes, showing their isomorphism types are determined by these classes and relating their properties to the underlying simplicial complex.

Contribution

It introduces a method to construct vector bundles with given characteristic classes over Davis-Januszkiewicz spaces and analyzes their structural properties and relations to quasitoric manifolds.

Findings

Constructed vector bundles with prescribed Chern, Pontrjagin, and Euler classes.

Isomorphism types are determined by characteristic classes.

Bundles relate closely to tangent bundles of quasitoric manifolds.

Abstract

For any (n-1)-dimensional simplicial complex, we construct a particular n-dimensional complex vector bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley Reisner algebra. We show that the isomorphism type of this complex vector bundle as well as of its realification are completely determined by its characteristic classes. This allows us to show that coloring properties of the simplicial complex are reflected by splitting properties of this bundle and vice versa. Similar question are also discussed for 2n-dimensional real vector bundles with particular prescribed characteristic Pontrjagin and Euler classes. We also analyze which of these bundles admit a complex structure. It turns out that all these bundles are closely related to the tangent bundles of quasitoric manifolds and moment angle complexes.