Equivariant Topology of Configuration Spaces

TL;DR

This paper investigates the equivariant topology of configuration spaces, providing complete and partial calculations of the Fadell-Husseini index for prime and prime power cases, leading to new proofs and extensions of classical theorems.

Contribution

It introduces new formulas and descriptions for the Fadell-Husseini index of configuration spaces, extending equivariant topology tools and solving conjectures for prime cases.

Findings

Complete determination of Index_{Z/p}(F(R^d,p);F_p)

Partial description of Index_{(Z/p)^k}(F(R^d,p^k);F_p)

New proof of the Nandakumar & Ramana Rao conjecture for primes

Abstract

We study the Fadell-Husseini index of the configuration space F(R^d,n) with respect to different subgroups of the symmetric group S_n. For p prime and d>0, we completely determine Index_{Z/p}(F(R^d,p);F_p) and partially describe Index{(Z/p)^k}(F(R^d,p^k);F_p). 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 Pi_p as an F_p[Z_p]-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^k a prime power, we compute the Lusternik-Schnirelmann category cat(F(R^d,n)/S_n)=(d-1)(n-1). Moreover, we extend coincidence results related to the Borsuk-Ulam theorem, as obtained by Cohen & Connett, Cohen & Lusk, and Karasev & Volovikov.