On subsets of $S^n$ whose $(n+1)$-point subsets are contained in open hemisheres

TL;DR

This paper studies special subsets of spheres where every three or fewer points lie in an open hemisphere, revealing polyhedral structures and providing examples, with implications for group cohomology.

Contribution

It demonstrates a natural polyhedral structure in certain sphere subsets satisfying a tameness condition related to the Bieri-Groves conjecture.

Findings

Polyhedrality in special sphere subsets.

Strong polyhedrality condition for open sets on the 2-sphere.

Includes numerous examples illustrating the concepts.

Abstract

We investigate the nature of subsets of spheres which satisfy a tameness condition associated with the Bieri-Groves conjecture on cohomological finiteness conditions for metabelian groups. We find that there is a natural polyhedrality in a crucial special case. In the case of the two dimensional sphere we establish a strong polyhedrality condition for certain open sets which are maximal subject to satisfying the tameness condition that subsets of three or fewer points are contained in open hemispheres. Many examples are included.