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Consequences of the kahn–priddy theorem in homotopy and geometry

Part of: Homotopy theory

Published online by Cambridge University Press:  26 February 2010

A. J. Berrick
A. J. Berrick
Affiliation:
Department of Mathematics, National University of Singapore, Kent Ridge, Singapore 0511.
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Extract

Much recent attention has been given to geometric representation of elements of the stable homotopy groups of spheres, π*s A particular example concerns non-singular bilinear maps ℝm+1 × ℝn+1 → ℝm+n+1−p; on restriction and normalisation these become biskew maps Sm × Sn ℝ Sm+n-p;. Now the Hopf construction ℋ applied to any map f: Sm × Sn → Sm+n-p yields

MSC classification

Secondary: 55P99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 28 , Issue 1 , June 1981 , pp. 72 - 78
Copyright
Copyright © University College London 1981

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References

1.Al-Sabti, G. and Bier, T.. “Elements in the stable homotopy group of spheres which are not bilinearly representable”, Bull. London Math. Soc, 10 (1978), 197–200.CrossRefGoogle Scholar
2.Berrick, A. J.. “Axial maps with futher structure”, Proc. Amer. Math. Soc, 54 (1976), 413–416.CrossRefGoogle Scholar
3.Berrick, A. J.. “Induction on symmetric axial maps and embeddings of projective spaces”, Proc. Amer. Math. Soc, 60 (1976), 276–278.CrossRefGoogle Scholar
4.Berrick, A. J.. “The Smale invariants of an immersed projective space”, Math. Proc. Camb. Phil. Soc, 86 (1979), 401–12.CrossRefGoogle Scholar
5.Berrick, A. J.. “Projective space immersions, bilinear maps and stable homotopy groups of spheres”, Proc Siegen Sympos. in Topology 1979, Springer Lecture Notes in Math., 788 (Springer, 1980), 1–22.Google Scholar
6.Boardman, J. M. and Steer, B.. “On Hopf invariants”, Comm. Math. Helv., 42 (1967), 180–221.CrossRefGoogle Scholar
7.Crabb, M. C. and Steer, B.. “Vector-bundle monomorphisms with finite singularities”, Proc. London Math. Soc. (3), 30 (1975), 1–39.CrossRefGoogle Scholar
8.Gray, B.. “Bilinear forms, I”, J. London Math. Soc. (2), 16 (1977), 124–130.CrossRefGoogle Scholar
9.Hoo, C. S. and Mahowald, M. E.. “Some homotopy groups of Stiefel manifolds”, Bull. Amer. Math. Soc, 71 (1965), 661–667.CrossRefGoogle Scholar
10.James, I. M.. “The Topology of Stiefel Manifolds”, London Math. Soc. Lecture Note Series, No. 24 (Cambridge Univ. Press, Cambridge, 1976).Google Scholar
11.Lam, K. Y.. “Non-singular bilinear maps and stable homotopy classes of spheres”, Math. Proc. Camb. Phil. Soc, 82 (1977), 419–425.CrossRefGoogle Scholar
12.Massey, W. S.. “On the normal bundle of a sphere imbedded in Euclidean space”, Proc. Amer. Math. Soc, 10 (1959), 959–964.CrossRefGoogle Scholar
13.Milgram, J.. “Immersing projective spaces”, Ann. of Math., 85 (1967), 473–182.CrossRefGoogle Scholar
14.Milgram, R. J., Strutt, J. and Zvengrowski, P.. “Computing projective stable stems with the Adams spectral sequence”. To appear.Google Scholar
15.Mukai, J.. “An application of the Kahn-Priddy theorem”, J. London Math. Soc. (2), 15 (1977), I 183–187.CrossRefGoogle Scholar
16.Raussen, M. and Smith, L.. “A geometric interpretation of sphere bundle boundaries and generalized J.-homomorphisms with an application to a diagram of I. M. James”, Quart. J. Math. Oxford (2 30 (1979), 113–117. IGoogle Scholar
17.Smale, S.. “The classification of immersions of spheres in euclidean spaces”, Ann. of Math. (2), 69 (1959), 327–344.CrossRefGoogle Scholar
18.Smith, L.. “Nonsingular bilinear forms, generalized J homomorphisms, and the homotopy of spheres i I”, Indiana Univ. Math. J., 27 (1978), 697–737.CrossRefGoogle Scholar