Equivariant topology of configuration spaces

Abstract

We study the Fadell-Husseini index of the configuration space F(Rd, n) with respect to various subgroups of the symmetric group Sn. For p prime and k ≥ 1, we compute IndexZ/p(F(Rd, p); Fp) and partially describe Index(Z/p)k (F(Rd, pk); 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 and Ramana Rao conjecture for primes. For n = pk a prime power, we compute the Lusternik-Schnirelmann category cat(F(Rd, n)/Sn) = (d - 1)(n - 1). Moreover, we extend coincidence results related to the Borsuk-Ulam theorem, as obtained by Cohen and Connett, Cohen and Lusk, and Karasev and Volovikov.