On the space of self homotopy equivalences of the projective plane
type: Item
instance of: scholarly article
title: On the space of self homotopy equivalences of the projective plane
zbMATH Open document ID: 0796.55003
author: Tsuneyo Yamanoshita
published in: Journal of the Mathematical Society of Japan
publication date: 1994-09-26T00:00:00Z
review text: Given a connected based CW complex \(X\), let \(G(X)\) and \(G_ 0(X)\) denote the spaces of self homotopy equivalences of \(X\) and of based self- homotopy equivalences of \(X\) respectively. For the projective plane \(p^ 2\) the author establishes a homeomorphism \(G(P^ 2) \cong \text{SO}(3) \times (G_ 0(P^ 2)/O(2))\) by showing that the following principal fibre bundles \[ \begin{aligned} \text{SO}(2) & \to G^ +(S^ 2) \to G_ 0^ +(S^ 2)/\text{SO}(2),\\ \text{SO}(2) & \to \text{map}_ 0(S^ 2,P^ 2;\pi)\to \text{map}_ 0(S^ 2,P^ 2; \pi)/\text{SO}(2),\\ \text{SO}(2) & \to G_ 0^ +(P^ 2) \to G_ 0^ +(P^ 2) /\text{SO}(2),\\ \text{SO}(3) & \to G(P^ 2) \to G_ 0(P^ 2) / \text{O}(2) \cong G_ 0^ +(P^ 2) / \text{SO}(2)\end{aligned} \] have cross-sections using the covering map \(\pi: S^ 2 \to P^ 2\), where \(G^ +\) denotes the path component in \(G\) of the identity map.
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 469074
zbMATH Keywords: self homotopy equivalences
zbMATH Keywords: projective plane
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04530489
OpenAlex ID: W2006423225
DOI: 10.2969/JMSJ/04530489
description: scientific article; zbMATH DE number 469074