Twisted homology of configuration spaces and homology of spaces of equivariant maps
type: Item
instance of: scholarly article
title: Twisted homology of configuration spaces and homology of spaces of equivariant maps
zbMATH Open document ID: 1412.55015
DOI: 10.1134/S106456241807030X
published in: Doklady Mathematics
publication date: 2019-03-15T00:00:00Z
review text: For a positive integer $N$ and a topological space $X$, let $I(X,N)$ denote the ordered configuration space of $N$ distinct points in $X$ defined by $I(X,N)=\{(x_1,\dots ,x_N)\in X^N:x_i\not= x_j\text{ if }i\not= j\}$. The symmetric group $S(N)$ of $N$-letters acts freely on $I(X,N)$ by permutation of coordinates and let $B(X,N)$ denote the unordered configuration space of $N$ distinct points in $X$ given by the orbit space $B(X,N)=I(X,N)/S(N)$. For a local system $L$ of groups on $X$, let $\overline{H}_*(X;L)$ denote the (Borel-Moore) homology group of $X$ with coefficients $L$. \par Let us consider the non-trivial local system of groups on the $m$ dimensional real projective space $\mathbb{R}\mathrm{P}^m$; the non-zero element of $\pi_1(\mathbb{R}\mathrm{P}^m)$ acts on the fibers as multiplication by $-1$. Let $\Theta$ denote the local system on the space $(\mathbb{R}\mathrm{P}^m)^N$ given by the tensor product of $n$ systems lifted by the standard projection from the above systems on the factors. \par Let $\pm \mathbb{Q}$ (resp. $\tilde{\Theta}$) denote the local system on $B(\mathbb{R}\mathrm{P}^m,N)$ with fiber $\mathbb{Q}$, which resolves over the loops defining odd permutations of $N$ points (resp. the loops such that the union of traces of all $N$ points during the movement along the loop defines the non-zero element of the group $H_1(\mathbb{R}\mathrm{P}^m;\mathbb{Z}/2)$). Note that the restriction of $\Theta$ to the subspace $I(\mathbb{R}\mathrm{P}^m,N)$ is isomorphic to the local system lifted from $\tilde{\Theta}$ by the standard covering $I(\mathbb{R}\mathrm{P}^m,N)\to B(\mathbb{R}\mathrm{P}^m,N)$. \par In this paper the author investigates the homology groups $\overline{H}_*(B(\mathbb{R}\mathrm{P}^m,N);L)$ $(L=\tilde{\Theta}$ or $\tilde{\Theta}\otimes \pm \mathbb{Q}$) and $H_*(I(\mathbb{R}\mathrm{P}^m,N);\Theta)$. In particular, when $m$ is odd, he computes the Poincaré series of the group $H_*(B(\mathbb{R}\mathrm{P}^m,N);L)$ explicitly and he shows th…
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55R80
Mathematics Subject Classification ID: 55S15
Mathematics Subject Classification ID: 55N25
zbMATH DE Number: 7036541
zbMATH Keywords: configuration space
zbMATH Keywords: simplicity resolution
zbMATH Keywords: stable homology
zbMATH Keywords: non-resultant polynomial
zbMATH Keywords: equivariant map
zbMATH Keywords: equivariant function space
author: V. A. Vassiliev
MaRDI profile type: Publication
cites work: A generalization of the Eilenberg-Moore spectral sequence
cites work: The topology of rational functions and divisors of surfaces
cites work: Topology of complements of discriminants and resultants
cites work: Simplicial resolutions and spaces of algebraic maps between real projective spaces
cites work: Q5543303
cites work: Q4004217
cites work: Rational homology of the order complex of zero sets of homogeneous quadratic polynomial systems in \(\mathbb{R}^3\)
cites work: Homology groups of spaces of non-resultant quadratic polynomial systems in \({\mathbb R}^3\)
cites work: Stable cohomology of spaces of non-resultant polynomial systems in \(\mathbb{R}^3\)
cites work: Stable cohomology of spaces of non-resultant systems of homogeneous polynomials in \(\mathbb{R}^n\)
cites work: Homology of the complex of not 2-divisible partitions
full work available at URL: s106456241807030x
OpenAlex ID: W2911018778
description: scientific article; zbMATH DE number 7036541