Order of the canonical vector bundle over configuration spaces of spheres
type: Item
instance of: scholarly article
title: Order of the canonical vector bundle over configuration spaces of spheres
zbMATH Open document ID: 1400.55008
author: Shiquan Ren
published in: Forum Mathematicum
publication date: 2018-10-04T00:00:00Z
review text: For a vector bundle \(\xi\) let \(\xi^{\oplus t}\) denote the \(t\)-fold Whitney sum with itself and let \(\epsilon^s\) denote the trivial vector bundle of dimension \(s\). If there exists a positive integer \(t\) such that \(\xi^{\oplus t}\) is trivial, \(\xi\) is said to have a finite order and the smallest such \(t\) is called the order of \(\xi\). Similarly, \(\xi\) is said to be stably trivial if there exist positive integers \(t\) and \(s\) such that \(\xi\oplus \epsilon^t\) is isomorphic to \(\epsilon^s\). If there exists a positive integer \(t\) such that \(\xi^{\oplus t}\) is stably trivial then \(\xi\) is said to have finite stable order and the smallest such \(t\) is called the stable order of \(\xi\). For a relative CW complex \((M,M_0)\) and a positive integer \(k\geq 2\), let \(F(M| M_0,k)\) denote the configuration space defined by \[ F(M| M_0,k)= \{(x_1,\dots ,x_k)\in M^k:x_i\not= x_j \text{ if }i\not= j \text{ and } x_i\in M_0 \text{ for some }i\} \] The symmetric group \(\Sigma_k\) of \(k\) letters acts on \(F(M| M_0,k)\) freely by the usual permutation of coordinates, and let \(F(M| M_0,k)/\Sigma_k\) denote its orbit space. Let \(\xi_{M| M_0,k}\) denote the \(k\)-dimensional real vector bundle given by \(F(M|M_0,k)\times_{\Sigma_k}\mathbb{R}^k\to F(M|M_0,k)/\Sigma_k\) and we write \(\xi_{M,k}\) when \(M=M_0\). For a positive integer \(t\) let \(\rho (t)\) denote the number of positive integers \(1\leq k\leq t\) such that \(k\equiv 0,1,2,4\) mod \(8\), and let \(a_{m,k}\) denote the positive integer defined by \(a_{m,k}=2^{\rho (m-1)}\prod_{3\leq p\leq k,p\text{ prime }}p^{\lfloor \frac{m-1}{2}\rfloor}.\) In this paper, the author studies the order and stable order of \(\xi_{M| M_0,k}\) when \(M=\coprod_nS^m\) and \(M_0\) is a non-empty CW-subcomplex of \(M\). More precisely, he generalizes the result obtained by \textit{F. R. Cohen} et al. [Pac. J. Math. 104, 47--54 (1983; Zbl 0513.55007)] and he proves that the order and the stable order of \(\xi…
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55R80
Mathematics Subject Classification ID: 55R10
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55P40
zbMATH DE Number: 6947023
zbMATH Keywords: vector bundle
zbMATH Keywords: configuration space
zbMATH Keywords: stable homotopy type
zbMATH Keywords: Whitney sum
zbMATH Keywords: order
zbMATH Keywords: stable order
MaRDI profile type: Publication
OpenAlex ID: W2801108015
cites work: Vector fields on spheres
cites work: Q3782446
cites work: On the homology of configuration spaces
cites work: Homological stability for configuration spaces of manifolds
cites work: Bundles over configuration spaces
cites work: Q4204807
cites work: Q4192675
cites work: The geometry of iterated loop spaces
cites work: Configuration spaces of positive and negative particles
cites work: ON THE COHOMOLOGY OF CONFIGURATION SPACES ON SURFACES
cites work: Configuration spaces on the sphere and higher loop spaces
cites work: The cohomology of projectively compactified complex swallow-tails and their complements
cites work: Order of the identity of the stable summands of \(\Omega^{2k}S^{2n+1}\)
cites work: On the homology of configuration spaces \(C((M, M_0) \times R^n; X)\)
cites work: Order of the canonical vector bundle on \(C_n(k)/\Sigma_k\)
Wikidata QID: Q129894128
DOI: 10.1515/FORUM-2018-0046
description: scientific article; zbMATH DE number 6947023