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Quadratic forms between spheres and the non-existence of sums of squares formulae

  • Paul Y. H. Yiu (a1)

Hurwitz [6] posed in 1898 the problem of determining, for given integers r and s, the least integer n, denoted by r s, for which there exists an [r, s, n] formula, namely a sums of squares formula of the type

where are bilinear forms with real coefficients in and . Such an [r, s, n] formula is equivalent to a normed bilinear map satisfying . We shall, therefore, speak of sums of squares formulae and normed bilinear maps interchangeably.

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[1] J. F. Adams . On the nonexistence of elements of Hopf invariant one. Ann. of Math. 72 (1960), 20104.

[3] G. Al-Sabti and T. Bier . Elements in the stable homotopy groups of spheres which are not bilinearly representable, Bull. London Math. Soc. 10 (1978), 197200.

[5] H. Hopf . Ein topologischer Beitrag zur reellen Algebra. Comment. Math. Helv. 13 (1941), 219239.

[7] A. Hurwitz . Über die Komposition der quadratischen Formen. Math. Ann. 88 (1923), 125; reprinted in Math. Werke Bd. 2, pp. 641–666.

[8] K. Y. Lam . Construction of nonsingular bilinear maps. Topology 6 (1967), 423426.

[12] K. Y. Lam . Some new results in composition of quadratic forms. Invent. Math. 79 (1985), 467474.

[13] R. J. Milgram . Immersing protective spaces. Ann. Math. 85 (1967), 473482.

[15] G. F. Paechter . The groups . Quart. J. Math. Oxford7 (1956), 249268.

[16] J. Radon . Lineare Scharen orthogonalen Matrizen. Abh. Math. Sem. Univ. Hamburg1 (1922), 114.

[17] J. Roitberg . Dilatation phenomena in the homotopy groups of spheres. Advances in Math. 15 (1975), 198206.

[19] E. Stiefel . Üjer Richtungsfelder in den projektiven Raumen und einen Satz aus reellen Algebra. Comment. Math. Helv. 13 (1941), 201218.

[21] R. Wood . Polynomial maps from spheres to spheres. Invent. Math. 5 (1968), 163168.

[24] S. Yuzvinsky . A series of monomial pairings. Linear and Multilinear Algebra 15 (1984), 109119.

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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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