Equivariant Topology of Configuration Spaces
نویسندگان
چکیده
We study the Fadell–Husseini index of the configuration space F (R, n) with respect to different subgroups of the symmetric group Sn. For p prime and k ≥ 1, we completely determine IndexZ/p(F (R, p);Fp) and partially describe Index(Z/p)k (F (R, p);Fp). In this process we obtain results of independent interest, including: (1) an extended equivariant Goresky–MacPherson formula, (2) a complete description of the top homology of the partition lattice Πp as an Fp[Zp]-module, and (3) a generalized Dold theorem for elementary abelian groups. The results on the Fadell–Husseini index yield a new proof of the Nandakumar & Ramana Rao conjecture for a prime. For n = p a prime power, we compute the Lusternik–Schnirelmann category cat(F (R, n)/Sn) = (d − 1)(n − 1), and for spheres obtain the bounds (d − 1)(n − 1) ≤ cat(F (S, n)/Sn) ≤ (d − 1)(n − 1) + 1. Moreover, we extend coincidence results related to the Borsuk–Ulam theorem, as obtained by Cohen & Connett, Cohen & Lusk, and Karasev & Volovikov.
منابع مشابه
Equivariant configuration spaces
We use the compression theorem (cf [7; section 7]) to prove results for equivariant configuration spaces analogous to the well-known non-equivariant results of May, Milgram and Segal [5,6,8]. AMS Classification 55P91, 55P35, 55P40; 57R91, 55P45, 55P47
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