Stable maps to projective spaces
type: Item
instance of: scholarly article
title: Stable maps to projective spaces
zbMATH Open document ID: 0657.55010
author: Kaoru Morisugi
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1988-01-01T00:00:00Z
review text: Let \(HP^{\infty}\) (resp. \(Q^{\infty})\) be the infinite dimensional quaternionic projective space (resp. the infinite dimensional quaternionic quasi-projective space). In this paper the author studies the natural homomorphism from \(\{\) X,Y\(\}\) to \(Hom(H_*(X)\), \(H_*(Y))\) when Y is \(HP^{\infty}\) or \(Q^{\infty}\), all homology groups have integer coefficients and where \(\{\) X,Y\(\}\) means the based homotopy set of stable maps from X to Y. In a previous paper [Publ. Res. Inst. Math. Sci. 20, 971-976 (1984; Zbl 0556.55007)] the author determined the image of the induced homomorphism from \(\{\Sigma^{4n}HP^{\infty}\), \(HP^{\infty}\}\) to \(Hom(H_*(HP^{\infty})\), \(H_*(HP^{\infty}))\). In that paper, for \(Q^{\infty}\) there was a mistake. The author says that this mistake was pointed out by \textit{H. Oshima} who proved a corrected theorem (unpublished). In this paper the author gives a slightly different (as the author declares) statement of this theorem. Then, using a similar method, the author constructs various stable maps between projective spaces so that the homology homomorphisms induced by those stable maps give the basis of the image from stable maps to their induced homology homomorphisms. The method used (that introduced in the op. cit. of the author) can be extended to cases where the source spaces of stable maps satisfy certain conditions.
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 55Q99
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 4073963
zbMATH Keywords: maps into projective spaces
zbMATH Keywords: infinite dimensional quaternionic projective space
zbMATH Keywords: infinite dimensional quaternionic quasi-projective space
zbMATH Keywords: homotopy set of stable maps
zbMATH Keywords: stable maps between projective spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195175204
OpenAlex ID: W2022093209
cites work: Q4057799
cites work: On the codegree of negative multiples of the Hopf bundle
cites work: Q4105006
cites work: Self Maps of Projective Spaces
cites work: On the structure of Hopf algebras
cites work: Stable self maps of the quaternionic (quasi-)projective space
cites work: THE STABLE HOMOTOPY OF COMPLEX PROJECTIVE SPACE
cites work: ESTIMATES FOR THE COMPLEX AND QUATERNIONIC JAMES NUMBERS
DOI: 10.2977/PRIMS/1195175204
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description: scientific article; zbMATH DE number 4073963