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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] Adams J. F.. On the groups J(X), IV. Topology 5 (1966), 2171.
[2] Adams J. F.. The Kahn-Priddy theorem. Math. Proc. Cambridge Philos. Soc. 73 (1973), 4555.
[3] Anderson D. W.. The e invariant and the Hopf invariant. Topology 9 (1970), 4954.
[4] Atiyah M. F.. Thorn complexes. Proc. London Math. Soc. (3) 11 (1961), 291310.
[5] Bendersky M.. Unstable towers in the mod p homotopy groups of spheres (in preparation) (1983).
[6] Cohen F. R., Moore J. C. and Neisendorfer J. A.. Decompositions of loop spaces and applications to exponents. Algebraic Topology, Aarhus 1978, Lecture Notes in Math, vol. 763 (Springer-Verlag, 1979), 112.
[7] Gray B.. The odd components of the unstable homotopy groups of spheres (mimeographed notes) (1967).
[8] Gray B.. On the sphere of origin of infinite families in the homotopy groups of spheres. Topology 8 (1969), 219232.
[9] Holzager R.. Stable splitting of K (G, 1), Proc. Amer. Math. Soc. 31 (1972), 305306.
[10] Kambe T.. The structure of KA-rings of the lens space and their applications. J. Math. Soc. Japan, 18 (1966), 135146.
[11] Kahn D. S. and Priddy S. B.. The transfer and stable homotopy theory. Math. Proc. Cambridge Philos. Soc. 83 (1978), 103111.
[12] Kambe T., Matsunage H. and Toda H.. A note on stunted lens space. J. Math. Kyoto Univ. 5 (2) (1966), 143149.
[13] Mahowald M.. The image of J in the EHP sequence. Annals of Math. 116 (1982), 65112.
[14] May J. P.. An algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications, Lecture Notes in Math. vol. 168 (Springer-Verlag, 1970), 153231.
[15] Moore J. C.. Course Lectures at Princeton University, Fall 1976.
[16] Selick P.. Odd primary torsion in πk(S3). Topology 17 (1978), 407412.
[17] Selick P.. A spectral sequence concerning the double suspension. Invent. Math. 64 (1981), 1524.
[18] Toda H.. p-primary components of the homotopy groups of spheies. IV. Compositions and toric constructions. Mem. Coll. Sci. Kyoto, Ser. A 32, (1959), 297332.
[19] Toda H.. On the double suspension E2. Journal of Inst. Poly. Osaka City University 7 (1956), 103145.
[20] Toda H.. On iterated suspensions. II. J. Math. Kyoto Univ., 5 (3) (1966), 209250.
[21] Toda H.. An important relation in homotopy groups of spheres. Proc. Japan Acad. 43 (1967), 839842.
[22] Wilkerson C.. 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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