TY - JOUR
T1 - Deformation retracts of neighborhood complexes of stable kneser graphs
AU - Braun, Benjamin
AU - Zeckner, Matthew
PY - 2014
Y1 - 2014
N2 - In 2003, A. Björner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SGn,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 SGn,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 SGn,2.
AB - In 2003, A. Björner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SGn,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 SGn,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 SGn,2.
KW - Discrete morse theory
KW - Neighborhood complex
KW - Polytope
KW - Stable kneser graph
UR - http://www.scopus.com/inward/record.url?scp=84889017783&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84889017783&partnerID=8YFLogxK
U2 - 10.1090/s0002-9939-2013-11803-4
DO - 10.1090/s0002-9939-2013-11803-4
M3 - Article
AN - SCOPUS:84889017783
SN - 0002-9939
VL - 142
SP - 413
EP - 427
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 2
ER -