Lower Bounds for Cubical Pseudomanifolds

TL;DR

This paper establishes vertex lower bounds for cubical pseudomanifolds and proves the generalized lower bound conjecture for all cubical 4-spheres and certain higher-dimensional classes using cubical h-vectors.

Contribution

It provides new lower bounds for vertices in cubical pseudomanifolds and confirms the generalized lower bound conjecture for specific cubical spheres.

Findings

Vertices in a d-dimensional cubical pseudomanifold are at least 2^{d+1}.

The generalized lower bound conjecture holds for all cubical 4-spheres.

It verifies the conjecture for some special classes of higher-dimensional cubical spheres.

Abstract

It is verified that the number of vertices in a $d$-dimensional cubical pseudomanifold is at least $2^{d+1}$. Using Adin's cubical $h$-vector, the generalized lower bound conjecture is established for all cubical 4-spheres, as well as for some special classes cubical spheres in higher dimensions.