Generalized 1-skeleta and a lifting result

TL;DR

This paper extends the theory of 1-skeleta to a broader class including projected simple polytopes and proves a lifting theorem that characterizes when a 1-skeleton arises from such polytopes.

Contribution

It generalizes existing notions of 1-skeleta and establishes a new lifting result for their characterization in relation to projected simple polytopes.

Findings

Extended the class of 1-skeleta to include projected simple polytopes

Proved a lifting theorem characterizing 1-skeleta from projected simple polytopes

Connected Morse theory techniques to the study of generalized 1-skeleta

Abstract

In their paper "1-skeleta, Betti numbers, and equivariant cohomology" Guillemin and Zara described some beautiful constructions enabling them to use Morse theory on a certain class 1-skeleta including 1-skeleta of simple polytopes. In this paper we extend some of the notions and constructions from that paper to a larger class of 1-skeleta that includes 1-skeleta of projected simple polytopes. As an application of these ideas we prove a lifting result for 1-skeleta, which yields a characterization of 1-skeleta coming from projected simple polytopes.