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Unstable families related to the image of J

  • Brayton Gray (a1)

The object of this paper is to describe certain families of unstable elements in the homotopy groups of spheres at an odd prime. In so doing we completely account for the image of J as possible Hopf invariants of unstable elements. The analogous result for p = 2 was obtained in [13]. In addition we will discuss other periodic phenomena. Our main results have been independently obtained by Bendersky[5] using BP*. Our methods, however, are entirely geometric, and we actually construct the elements, rather than detect them. Our basic tool is the map .All our constructions are made in BΣp and transferred over.

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[1] J. F. Adams . On the groups J(X), IV. Topology 5 (1966), 2171.

[3] D. W. Anderson . The e invariant and the Hopf invariant. Topology 9 (1970), 4954.

[8] B. Gray . On the sphere of origin of infinite families in the homotopy groups of spheres. Topology 8 (1969), 219232.

[9] R. Holzager . Stable splitting of K (G, 1), Proc. Amer. Math. Soc. 31 (1972), 305306.

[10] T. Kambe . The structure of KA-rings of the lens space and their applications. J. Math. Soc. Japan, 18 (1966), 135146.

[13] M. Mahowald . The image of J in the EHP sequence. Annals of Math. 116 (1982), 65112.

[16] P. Selick . Odd primary torsion in πk(S3). Topology 17 (1978), 407412.

[17] P. Selick . A spectral sequence concerning the double suspension. Invent. Math. 64 (1981), 1524.

[21] H. Toda . An important relation in homotopy groups of spheres. Proc. Japan Acad. 43 (1967), 839842.

[22] C. Wilkerson . Genus and cancelation. Topology 14 (1975), 2936.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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