Constructions of $d$-spheres from $(d-1)$-spheres and $d$-balls with same set of vertices

TL;DR

This paper investigates vertex-preserving constructions of higher-dimensional spheres from lower-dimensional spheres and balls, providing affirmative answers for specific classes like flag, stacked, and join spheres.

Contribution

It demonstrates that certain classes of spheres can be constructed without additional vertices, advancing understanding of combinatorial sphere constructions.

Findings

Vertex-preserving constructions are possible for flag, stacked, and join spheres.

The paper extends the theory of combinatorial sphere construction methods.

It explores the existence of n-vertex spheres containing given n-vertex balls.

Abstract

Given a combinatorial $(d-1)$-sphere $S$, to construct a combinatorial $d$-sphere $S^{\hspace{.2mm}\prime}$ containing $S$, one usually needs some more vertices. Here we consider the question whether we can do one such construction without the help of any additional vertices. We show that this question has affirmative answer when $S$ is a flag sphere, a stacked sphere or a join of spheres. We also consider the question whether we can construct an $n$-vertex combinatorial $d$-sphere containing a given $n$-vertex combinatorial $d$-ball.