Equivariant topology of configuration spaces. (21st March 2015)

• Record Type: Journal Article
• Authors: Blagojević, Pavle V. M.; Lück, Wolfgang; Ziegler, Günter M.
• Is Part Of: Journal of topology. Volume 8:Part 2(2015)
• Page Start: 414
• Page End: 456
• Publication Date: 2015-03-21
• Subjects: Topology -- Periodicals; 514.05
• DOI: 10.1112/jtopol/jtv002
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract : We study the Fadell–Husseini index of the configuration spaceF ( R d, n ) with respect to various subgroups of the symmetric groupS n . For p prime andk ⩾ 1, we computeIndex Z / p ( F ( R d, p ) ; F p ) and partially describeIndex ( 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 latticeΠ p as anF 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 and Ramana Rao conjecture for primes. Forn = p k a prime power, we compute the Lusternik–Schnirelmann categoryCat ( 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 and Connett, Cohen and Lusk, and Karasev and Volovikov.