2023/03/21 by Okura, Kengo · 1 citation
#05C69 #05C76 #05E45 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2303.11984
We consider a certain class of simplicial complexes which includes the independence complexes of forests. We show that if a simplicial complex K belongs to this class, then the polyhedral join Z^*K(\underlineX, ∅) is homotopy equivalent to a wedge sum of CW complexes of the form Σr Xi1 * Xi2 * ⋯ * Xik, where \underlineX is a family \Xi\i ∈ V(K) of CW complexes and Σ denotes the unreduced suspension. This result is applied to study the homotopy type of the independence complex of the lexicographic product G[H] of a graph H over a forest G. We denote by Lm a tree on m vertices with no branches. We show that the geometric realization of the independence complex of Lm [H] is homotopy equivalent to a wedge sum of spheres if m ≠ 2,3 and the geometric realization of the independence complex of H is homotopy equivalent to a wedge sum of same dimensional spheres.