The stable Kneser graph SGn,k​, n≥1, k≥0, introduced by Schrijver
\cite{schrijver}, is a vertex critical graph with chromatic number k+2, its
vertices are certain subsets of a set of cardinality m=2n+k. Bj\"orner and de
Longueville \cite{anders-mark} have shown that its box complex is homotopy
equivalent to a sphere, \Hom(K_2,SG_{n,k})\homot\Sphere^k. 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(K_2,SG_{n,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(SG_{2s,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(SG_{n,k}, G) is (r−1)-connected and χ(G)<r+k+2.Comment: 34 pp