Self maps of \(\Sigma ^ k{\mathbb{C}}P^ 3\) for k\(\geq 1\)
type: Item
instance of: scholarly article
title: Self maps of \(\Sigma ^ k{\mathbb{C}}P^ 3\) for k\(\geq 1\)
zbMATH Open document ID: 0729.55006
author: Kohhei Yamaguchi
published in: Kodai Mathematical Journal
publication date: 1991-01-01T00:00:00Z
review text: If X is a pointed space, then \(M(X)=[X,X]\), the set of (based) homotopy classes of self-maps of X, becomes a monoid under multiplication. If, in addition, X is a co-H-space, then M(X) is a ring. Particularly interesting is the group E(X) of units in M(X). In the paper under review, the group E(X) resp. the ring M(X) are explicitly calculated for the suspension \(\Sigma^ k{\mathbb{C}}P^ 3\) of the complex projective 3-space, \(k\geq 1\). The descriptions of E(X) and M(X) are in terms of generators and relations, the formulae are too complicated to be presented here. As a corollary of his results, the author is able to describe precisely those endomorphisms of \(H_*(\Sigma^ k{\mathbb{C}}P^ 3)\) which are realized by a continuous self-map. In the proof the author calculates the homotopy groups \(\pi_ q(\Sigma^ k{\mathbb{C}}P^ 3)\) in a certain range of dimensions.
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55Q52
zbMATH DE Number: 4205396
zbMATH Keywords: homotopy classes
zbMATH Keywords: self-maps
zbMATH Keywords: suspension
zbMATH Keywords: complex projective 3-space
zbMATH Keywords: homotopy groups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039344
OpenAlex ID: W2095306581
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DOI: 10.2996/KMJ/1138039344
description: scientific article; zbMATH DE number 4205396