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Homotopy of the exceptional Lie group \(G_ 2\). (English) Zbl 0604.55005

Let \(M^ n\) be the mod 2 Moore space \(S^ n \cup_{2\iota} e^{n+1}\), where \(\iota\) is the generator of \(\pi^ S_ 0(S^ 0)\), and let \(S^ n\to^{i}M^ n\to^{p}S^{n+1}\) be the natural cofibration. Define \({\bar \eta}\): \(M^{n+1}\to S^ n\) and \({\tilde \eta}\): \(S^{n-1}\to M^{n-3}\) to be maps such that \({\bar \eta}\)i\(=\eta\) and p\({\tilde \eta}=\eta\), where \(\eta\) is the generator of \(\pi^ S_ 1(S^ 0)\), and define complexes \(X^ n\) (n\(\geq 3)\) and \(Y^ n\) (n\(\geq 6)\) as the mapping cones of \({\bar \eta}\) and \({\tilde \eta}\), respectively. The author proves that \(X^ n\) and \(Y^ n\) are K-theory spheres, obtains a fibration \(X^ 3\to Q\to Y^{11}\), where Q is the space such that there is the stable decomposition of \(G_ 2\) as \(Q\vee S^ d\), and computes the stable class of the attaching map \(\phi\) : \(Y^{11}\to \Sigma X^ 3\) by using the complex Adams e-invariant \(e: \{\) \(Y^{11},\Sigma X^ 3\}\to {\mathbb{Q}}/ {\mathbb{Z}}.\)
The main results are as follows: Theorem A. \(\{Y^{11},\Sigma X^ 3\}={\mathbb{Z}}/60\) with generator of e-invariant 1/60\(\in {\mathbb{Q}}/ {\mathbb{Z}}\). Theorem B. Stably the attaching map \(\phi\) is twice a generator, and hence, is of order 30. Furthermore the author has results on self-maps of \(G_ 2\). (We were saddened to hear that the author died suddenly on 30 October 1984. Dr. M. Crabb has kindly revised this paper in accordance with the referee’s suggestions.) In memory of Professor Shichiro Oka,
Reviewer: T.Kobayashi

MSC:

55Q52 Homotopy groups of special spaces
55Q10 Stable homotopy groups
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
55Q45 Stable homotopy of spheres
55P45 \(H\)-spaces and duals
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References:

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