The equivariant topology of stable Kneser graphs

The stable Kneser graph SGn,k, n⩾1, k⩾0, introduced by Schrijver (1978) [19], , is a vertex critical graph with chromatic number k+2, its vertices are certain subsets of a set of cardinality m=2n+k. Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom(K2,SGn,k)≃Sk. The dihedral group D2m acts canonically on SGn,k, the group C2 with 2 elements acts on K2. We almost determine the (C2×D2m)-homotopy type of Hom(K2,SGn,k) and use this to prove the following results.The graphs SG2s,4 are homotopy test graphs, i.e. for every graph H and r⩾0 such that Hom(SG2s,4,H) is (r−1)-connected, the chromatic number χ(H) is at least r+6.If k∉{0,1,2,4,8} and n⩾N(k) then SGn,k is not a homotopy test graph, i.e. there are a graph G and an r⩾1 such that Hom(SGn,k,G) is (r−1)-connected and χ(G)