On the stable homotopy of the real projective space of even low dimension
type: Item
instance of: scholarly article
title: On the stable homotopy of the real projective space of even low dimension
zbMATH Open document ID: 0605.55012
author: Juno Mukai
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1986-01-01T00:00:00Z
review text: The purpose of this paper is to determine the group structure of the stable homotopy group \(\{P^{2n},P^{2n}\}\) for \(n\leq 4\), where \(P^ n\) denotes the real n-dimensional projective space. As a corollary, the stable group of self-homotopy equivalences of \(P^{2n}\) for \(n\leq 4\) is determined. The first motivation of this work is to ascertain Adams' theorem about the uniqueness of the existence of the Kahn-Priddy map. Our method is to use the composition methods developed by Toda. In particular, we use the knowledge of the following: The order of the identity class of \(P^{2n}\), the order of the Kahn-Priddy map and the ring structure of the stable homotopy ring of spheres in low dimension.
Mathematics Subject Classification ID: 55Q10
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55P10
zbMATH DE Number: 3977929
zbMATH Keywords: secondary composition
zbMATH Keywords: stable homotopy group
zbMATH Keywords: real n-dimensional projective space
zbMATH Keywords: group of self-homotopy equivalences
zbMATH Keywords: Kahn-Priddy map
zbMATH Keywords: composition methods
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195178373
OpenAlex ID: W2047098951
cites work: Thom Complexes
cites work: A Theorem in Homological Algebra and Stable Homotopy of Projective Spaces
cites work: The metastable homotopy of 𝑆ⁿ
cites work: STABLE HOMOTOPY OF SOME ELEMENTARY COMPLEXES
cites work: Homotopy of the exceptional Lie group G2
cites work: A remark on James numbers of Stiefel manifolds
cites work: Compositional Methods in Homotopy Groups of Spheres. (AM-49)
cites work: Order of the identity class of a suspension space
DOI: 10.2977/PRIMS/1195178373
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description: scientific article; zbMATH DE number 3977929