Non-projectability of polytope skeleta

TL;DR

This paper explores the limitations of projecting polytopes while preserving their k-skeleta, using topological combinatorics to identify obstructions in various polytope products.

Contribution

It develops a general framework to compute obstructions to polytope projections that preserve skeleta, extending previous work with new topological and combinatorial tools.

Findings

Identifies obstructions for projections of product polytopes

Calculates specific obstructions for polygons and simplices

Shows limitations of certain polytope constructions

Abstract

We investigate necessary conditions for the existence of projections of polytopes that preserve full k-skeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to e-space strictly preserves the k-skeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of graph colorings and linear integer programs. We focus on polytopes of product type and calculate the obstructions for products of polygons, products of simplices, and wedge products of polytopes. Our results show the limitations of constructions for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge product surfaces of R\"orig & Ziegler (2009) and complement their results.