Simple $S_r$-homotopy types of Hom complexes and box complexes associated to $r$-graphs

Details Simple $S_r$-homotopy types of Hom complexes and box complexes associated to $r$-graphs Simple $S_r$-homotopy types of Hom complexes and box complexes associated to $r$-graphs

2010-04-05

arxiv.org/abs/1004.0583v2

Mathematics

Algebraic Topology

12 pages, 1 figure

Scientific paper

For a pair $(H_1,H_2)$ of graphs, Lov\'{a}sz introduced a polytopal complex called the Hom complex $\text{Hom}(H_1,H_2)$, in order to estimate topological lower bounds for chromatic numbers of graphs. The definition is generalized to hypergraphs. Denoted by $K_r^r$ the complete $r$-graph on $r$ vertices. Given an $r$-graph $H$, we compare $\text{Hom}(K_r^r,H)$ with the box complex $\mathsf{B}_{\text{edge}}(H)$, invented by Alon, Frankl and Lov\'{a}sz. We verify that $\text{Hom}(K_r^r,H)$ and $\mathsf{B}_{\text{edge}}(H)$, both are equipped with right actions of the symmetric group on $r$ letters $S_r$, are of the same simple $S_r$-homotopy type.

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