Oka, Shichiro Homotopy of the exceptional Lie group \(G_ 2\). (English) Zbl 0604.55005 Proc. Edinb. Math. Soc., II. Ser. 29, 145-169 (1986). 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 Cited in 1 ReviewCited in 4 Documents 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 Keywords:exceptional Lie group; H-structure; mod 2 Moore space; K-theory spheres; stable decomposition of \(G_ 2\); Adams e-invariant; self-maps of \(G_ 2\) PDFBibTeX XMLCite \textit{S. Oka}, Proc. Edinb. Math. Soc., II. Ser. 29, 145--169 (1986; Zbl 0604.55005) Full Text: DOI References: [1] DOI: 10.1016/0040-9383(71)90017-6 · Zbl 0223.55029 [2] Yokota, Groups and representations (in Japanese) (1973) [3] Sawashita, Hiroshima Math. J. 14 pp 75– (1984) [4] DOI: 10.1016/0040-9383(67)90010-9 · Zbl 0186.57103 [5] DOI: 10.1093/qmath/35.2.115 · Zbl 0555.55010 [6] Browder, Pacific J. Math. 12 pp 411– (1962) · Zbl 0112.14502 [7] DOI: 10.2307/2372843 · Zbl 0101.39702 [8] DOI: 10.2307/2372574 · Zbl 0056.16401 [9] DOI: 10.1016/0040-9383(65)90051-0 · Zbl 0136.21001 [10] DOI: 10.1016/0040-9383(62)90107-6 · Zbl 0108.17801 [11] DOI: 10.1016/0040-9383(66)90004-8 · Zbl 0145.19902 [12] DOI: 10.2307/1970213 · Zbl 0112.38102 [13] Oka, Proceedings of the Northwestern Homotopy Theory Conference pp 267– (1983) · Zbl 0543.55009 [14] Oka, Mem. Fac. Sci. Kyushu Univ. Ser. A 35 pp 307– (1981) [15] Mukai, Mem. Fac. Sci. Kyushu Univ. Ser. A 20 pp 266– (1966) [16] Mimura, J. Math. Kyoto Univ. 3 pp 217– (1964) [17] Mimur, J. Math. Kyoto Univ. 21 pp 331– (1981) [18] Mimura, J. Math. Kyoto Univ. 13 pp 611– (1973) [19] Mimura, J. Math. Kyoto Univ. 6 pp 131– (1967) [20] Maruyama, Mem. Fac. Sci. Kyushu Univ. Ser. A 35 pp 375– (1981) [21] Maruyama, Mem. Fac. Sci. Kyushu Univ. Ser. A 38 pp 5– (1984) [22] Toda, Composition methods in homotopy groups of spheres (1962) · Zbl 0101.40703 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.