Deformation Retracts of Neighborhood Complexes of Stable Kneser Graphs

TL;DR

This paper investigates the topological structure of neighborhood complexes of stable Kneser graphs, proving they contain deformation retracts onto polyhedral boundary spheres for the case k=2, and explores their symmetry properties.

Contribution

It provides a positive answer to whether these complexes contain deformation retracts of polytope boundaries specifically for k=2, extending previous topological results.

Findings

Neighborhood complexes of SG_{n,2} deformation retract onto polyhedral boundary spheres.

Partially subdivided complexes can be retracted onto invariant subcomplexes.

The resulting boundary spheres are symmetric under automorphisms of SG_{n,2}.

Abstract

In 2003, A. Bjorner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SG_{n,k} is homotopy equivalent to a k-sphere. Further, for n=2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SG_{n,k} contains as a deformation retract the boundary complex of a simplicial polytope. Our purpose is to give a positive answer to this question in the case k=2. We also find in this case that, after partially subdividing the neighborhood complex, the resulting complex deformation retracts onto a subcomplex arising as a polyhedral boundary sphere that is invariant under the action induced by the automorphism group of SG_{n,2}.