Characteristic numbers of canonical toric manifolds and their applications

TL;DR

This paper computes characteristic numbers for canonical toric manifolds and their real versions, providing combinatorial and geometric insights into their bordism classes and immersion properties.

Contribution

It introduces explicit formulas for characteristic numbers of canonical toric manifolds and applies these to classify bordism classes and analyze embeddings.

Findings

Computed all Chern, Milnor, Pontryagin numbers for canonical toric manifolds.

Provided combinatorial criteria for bounding and immersion dimension estimates.

Identified new representatives of unitary bordism ring generators.

Abstract

We compute all the Chern, Milnor and Pontryagin numbers for canonical toric manifolds associated with abstract simplicial complexes and the Stiefel-Whitney numbers for their real counterparts. Applications include combinatorial characterizations of the unitary, oriented and unoriented bordism classes, new geometrical representatives of the unitary bordism ring generators, a combinatorial criterion for a canonical toric manifold to bound, as well as the dimension estimates for their immersions into euclidean spaces.