Shellability and Sphericity of the quasi-arc complex of the M\"obius strip

TL;DR

This paper proves the shellability of the quasi-arc complex of the M"obius strip, establishing its topological structure as a PL-sphere, and provides elementary proofs for related complexes.

Contribution

It introduces the first proof of shellability for the quasi-arc complex of the M"obius strip and offers elementary proofs for arc complexes of polygons and cylinders.

Findings

The quasi-arc complex of the M"obius strip is shellable.

Arc complexes of polygons and cylinders are shellable.

These complexes are topologically PL-spheres.

Abstract

Shellability of a simplicial complex has many useful structural implications. In particular, it was shown by Danaraj and Klee that every shellable pseudo-manifold is a PL-sphere. The purpose of this paper is to prove the shellability of the quasi-arc complex of the M\"obius strip. Along the way we provide elementary proofs of the shellability of the arc complex of the $n$-gon and the cylinder. In turn, applying the result of Danaraj and Klee, we obtain the sphericity of all of these complexes.