Evaluation fibrations and path-components of the mapping space \(M(\mathbb{S}^{n+k},\mathbb{S}^n)\) for \(8\leq k\leq 13\)
type: Item
instance of: scholarly article
title: Evaluation fibrations and path-components of the mapping space \(M(\mathbb{S}^{n+k},\mathbb{S}^n)\) for \(8\leq k\leq 13\)
zbMATH Open document ID: 1328.55011
author: Thiago de Melo
author: Marek Golasiński
published in: Ukrainian Mathematical Journal
publication date: 2015-11-13T00:00:00Z
review text: Let \(M(\mathbb{S}^{n+k},\mathbb{S}^n)\) denote the space of all continuous maps from the \((n+k)\)-sphere \(\mathbb{S}^{n+k}\) into the \(n\)-sphere \(\mathbb{S}^n\) with the compact open topology. For an element \(\alpha\in \pi_{n+k}(\mathbb{S}^n)\), let \(M_{\alpha}(\mathbb{S}^{n+k},\mathbb{S}^n)\) be its path-component containing \(\alpha\). In this paper, the authors consider the problem to estimate the number of homotopy types of path-components \(M_{\alpha}(\mathbb{S}^{n+k},\mathbb{S}^n)\) and their corresponding fibre homotopy types. In particular, they estimate the number of homotopy types of \(M_{\alpha}(\mathbb{S}^{n+k},\mathbb{S}^n)\) explicitly for \(8\leq k\leq 13\). Moreover, they also improve the number of strong homotopy types of the evaluation map \(\omega_{\alpha}:M_{\alpha}(\mathbb{S}^{n+k\mathbb{S}^n})\to \mathbb{S}^n\) for \(8\leq k\leq 13\).
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55R05
zbMATH DE Number: 6508259
zbMATH Keywords: mapping space
zbMATH Keywords: evaluation fibration
zbMATH Keywords: homotopy equivalence
zbMATH Keywords: Gottlieb group
zbMATH Keywords: path-component
zbMATH Keywords: fibrewise homotopy
MaRDI profile type: Publication
full work available at URL: s11253-014-0849-3
OpenAlex ID: W2016277357
cites work: On the Homotopy Classification of the Extensions of a Fixed Map
cites work: Q3274680
cites work: Basic constructions in rational homotopy theory of function spaces
cites work: Gottlieb groups of spheres
cites work: Evaluation fibrations and path-components of the map space \(M(\mathbb S^{n+k},\mathbb S^n)\) for \(0\leq k\leq 7\).
cites work: Coincidences of fibrewise maps between sphere bundles over the circle
cites work: A Certain Subgroup of the Fundamental Group
cites work: Evaluation Subgroups of Homotopy Groups
cites work: Equivalence of evaluation fibrations
cites work: THE HOMOTOPY PROBLEM FOR THE COMPONENTS IN THE SPACE OF MAPS ON THE n-SPHERE
cites work: Q4819371
cites work: Q3261851
cites work: Note on the Homotopy Properties of the Components of the Mapping Space X s p
cites work: Q5494064
cites work: Criteria for components of a function space to be homotopy equivalent
cites work: The homotopy groups of Lie groups of low rank
cites work: Compositional Methods in Homotopy Groups of Spheres. (AM-49)
cites work: On products in homotopy groups
cites work: Q4192669
DOI: 10.1007/S11253-014-0849-3
description: scientific article; zbMATH DE number 6508259