Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T09:27:43.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Unstable families related to the image of J

Published online by Cambridge University Press:  24 October 2008

Brayton Gray
Brayton Gray
Affiliation:
University of Illinois at Chicago
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 1 , July 1984 , pp. 95 - 113
Copyright
Copyright © Cambridge Philosophical Society 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adams, J. F.. On the groups J(X), IV. Topology 5 (1966), 21–71.CrossRefGoogle Scholar
[2] Adams, J. F.. The Kahn-Priddy theorem. Math. Proc. Cambridge Philos. Soc. 73 (1973), 45–55.CrossRefGoogle Scholar
[3] Anderson, D. W.. The e invariant and the Hopf invariant. Topology 9 (1970), 49–54.CrossRefGoogle Scholar
[4] Atiyah, M. F.. Thorn complexes. Proc. London Math. Soc. (3) 11 (1961), 291–310.CrossRefGoogle Scholar
[5] Bendersky, M.. Unstable towers in the mod p homotopy groups of spheres (in preparation) (1983).Google Scholar
[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), 1–12.Google Scholar
[7] Gray, B.. The odd components of the unstable homotopy groups of spheres (mimeographed notes) (1967).Google Scholar
[8] Gray, B.. On the sphere of origin of infinite families in the homotopy groups of spheres. Topology 8 (1969), 219–232.CrossRefGoogle Scholar
[9] Holzager, R.. Stable splitting of K (G, 1), Proc. Amer. Math. Soc. 31 (1972), 305–306.Google Scholar
[10] Kambe, T.. The structure of KA-rings of the lens space and their applications. J. Math. Soc. Japan, 18 (1966), 135–146.Google Scholar
[11] Kahn, D. S. and Priddy, S. B.. The transfer and stable homotopy theory. Math. Proc. Cambridge Philos. Soc. 83 (1978), 103–111.CrossRefGoogle Scholar
[12] Kambe, T., Matsunage, H. and Toda, H.. A note on stunted lens space. J. Math. Kyoto Univ. 5 (2) (1966), 143–149.Google Scholar
[13] Mahowald, M.. The image of J in the EHP sequence. Annals of Math. 116 (1982), 65–112.CrossRefGoogle Scholar
[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), 153–231.Google Scholar
[15] Moore, J. C.. Course Lectures at Princeton University, Fall 1976.Google Scholar
[16] Selick, P.. Odd primary torsion in πk(S3). Topology 17 (1978), 407–412.CrossRefGoogle Scholar
[17] Selick, P.. A spectral sequence concerning the double suspension. Invent. Math. 64 (1981), 15–24.CrossRefGoogle Scholar
[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), 297–332.Google Scholar
[19] Toda, H.. On the double suspension E2. Journal of Inst. Poly. Osaka City University 7 (1956), 103–145.Google Scholar
[20] Toda, H.. On iterated suspensions. II. J. Math. Kyoto Univ., 5 (3) (1966), 209–250.Google Scholar
[21] Toda, H.. An important relation in homotopy groups of spheres. Proc. Japan Acad. 43 (1967), 839–842.Google Scholar
[22] Wilkerson, C.. Genus and cancelation. Topology 14 (1975), 29–36.CrossRefGoogle Scholar