The genus and the category of configuration spaces
type: Item
instance of: scholarly article
title: The genus and the category of configuration spaces
zbMATH Open document ID: 1175.57026
published in: Topology and its Applications
publication date: 2009-08-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.0555
review text: If \(X\) is a topological space, we write \(H^*(X,M) = H^*(X\times EG/G;M)\) for a \(\mathbb Z[G]\)-module \(M\) (action by the fundamental group). \(H^*_G(M) = H^*_G(pt;M)\); the map \(X\to pt\) induces \(H^*_G(M)\to H^*_G(X,M)\). The upper cohomological index \(\overline{\text{ind}}_M X\) is the largest \(n\) for which this map is non-trivial (which may be \(\infty\)). A more complex numerical index is \(i_G(X)\), defined by \textit{A. Yu. Volovikov} [Mat. Sb. 191, No.~9, 3--22 (2000; Zbl 0987.57016)]. Let \(G = (\mathbb Z_p)^k\) be the elementary abelian group and \(X\) be a \(G\)-space. Suppose \(H^m(G,\mathbb Z_p), m<n\) are finite \(\mathbb Z_p[G]\) modules which are induced from proper subgroups of \(G\). Then \(i_G(X)\geq n+1\). This is applied to configuration spaces. The author also studies \(G\)-category of \(X\) [see \textit{T. Bartsch}, Topological methods for variational problems with symmetries. Lecture Notes in Mathematics. 1560. Berlin: Springer-Verlag (1993; Zbl 0789.58001)]. The \(G\)-category is estimated from below in terms of the distinct sets of orbits. This is applied to configuration spaces built from points in a manifold.
reviewed by: Donald W. Kahn
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 55R80
Mathematics Subject Classification ID: 55M30
zbMATH DE Number: 5597518
zbMATH Keywords: configuration space
zbMATH Keywords: equivariant topology
zbMATH Keywords: Lyusternik-Schnirelmann category
author: Roman Karasev
MaRDI profile type: Publication
OpenAlex ID: W2042205874
arXiv ID: 0905.0555
cites work: Q5836930
cites work: Q5812466
cites work: Q3245787
cites work: Cohomologies of the group COS mod 2
cites work: Q3866809
cites work: Configuration-Like Spaces and the Borsuk-Ulam Theorem
cites work: Q3708977
cites work: Invariants of the Lusternik-Schnirelmann Type and the Topology of Critical Sets
cites work: On the topology of algorithms. I
cites work: Braid group cohomologies and algorithm complexity
cites work: THE KNASTER PROBLEM AND ALMOST SPHERICAL SECTIONS
cites work: Q4039743
cites work: Q3209889
cites work: A THEOREM OF BOURGIN-YANG TYPE FOR $ \mathbb{Z}_p^n$-ACTION
cites work: Topological methods for variational problems with symmetries
cites work: Topology of cylic configuration spaces and periodic trajectories of multi-dimensional billiards
cites work: Topology of billiard problems. II
cites work: On the index of \(G\)-spaces
cites work: Q4520853
cites work: Q4528987
cites work: Topology of plane arrangements and their complements
cites work: Coincidence points of maps of $ \Bbb Z_p^n$-spaces
cites work: On the Cohen-Lusk theorem
cites work: On the category of Euclidean configuration spaces and associated fibrations
DOI: 10.1016/J.TOPOL.2009.06.012
description: scientific article