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    The equivariant topology of stable Kneser graphs

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    The stable Kneser graph SGn,kSG_{n,k}, nβ‰₯1n\ge1, kβ‰₯0k\ge0, introduced by Schrijver \cite{schrijver}, is a vertex critical graph with chromatic number k+2k+2, its vertices are certain subsets of a set of cardinality m=2n+km=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 D2mD_{2m} acts canonically on SGn,kSG_{n,k}, the group C2C_2 with 2 elements acts on K2K_2. We almost determine the (C2Γ—D2m)(C_2\times D_{2m})-homotopy type of \Hom(K_2,SG_{n,k}) and use this to prove the following results. The graphs SG2s,4SG_{2s,4} are homotopy test graphs, i.e. for every graph HH and rβ‰₯0r\ge0 such that \Hom(SG_{2s,4},H) is (rβˆ’1)(r-1)-connected, the chromatic number Ο‡(H)\chi(H) is at least r+6r+6. If kβˆ‰{ 0,1,2,4,8 }k\notin\set{0,1,2,4,8} and nβ‰₯N(k)n\ge N(k) then SGn,kSG_{n,k} is not a homotopy test graph, i.e.\ there are a graph GG and an rβ‰₯1r\ge1 such that \Hom(SG_{n,k}, G) is (rβˆ’1)(r-1)-connected and Ο‡(G)<r+k+2\chi(G)<r+k+2.Comment: 34 pp
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