Quasitoric manifolds and Small covers over properly colored polytopes: Immersions and Embeddings

TL;DR

This paper constructs specific small covers and quasitoric manifolds over properly colored polytopes, calculating their Stiefel-Whitney classes to determine immersion and embedding obstructions, especially for dimensions that are powers of two.

Contribution

It introduces a method to construct small covers and quasitoric manifolds with proper facet colorings and computes their Stiefel-Whitney classes to analyze immersion and embedding properties.

Findings

Calculated Stiefel-Whitney classes as obstructions

Determined maximum embedding dimensions for certain manifolds

Identified special cases where dimension bounds are achieved

Abstract

We construct small covers and quasitoric manifolds over $n$-dimensional simple polytopes which allow proper colorings of facets with $n$ colors. We calculate Stiefel-Whitney classes of these manifolds as obstructions to immersions and embeddings into Euclidean spaces. The largest dimension required for embedding is achieved in the case $n$ is a power of two.