Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-23T05:14:09.961Z Has data issue: false hasContentIssue false
  • English
  • Français

On a question of Milnor concerning singularities of maps

Published online by Cambridge University Press:  20 January 2009

Elmer G. Rees
Elmer G. Rees
Affiliation:
Department of Mathematics and Statistics, James Clerk Maxwell Building, King's Buildings, Edinburgh EH9 3JZ, UK
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Restrictions are given on the dimensions m and k for which there is a map f: ℝ m → ℝk whose Jacobian has rank k in a neighbourhood of a singular point if f is either quadratic or even. The restrictions are shown to be best possible in the quadratic case.

Keywords

singularitiesreal polynomialsMilnor's hypothesisnon-singular real matrices
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 1 , February 2000 , pp. 149 - 153
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1.Adams, J. F., Vector fields on spheres, Ann. Math. 75 (1962), 603–632.CrossRefGoogle Scholar
2.Adams, J. F., Lax, P. and Phillips, R., On matrices whose real linear combinations are non-singular, Proc. Am. Math. Soc. 16 (1965), 318–322.Google Scholar
3.Davis, D. M., Desuspensions of stunted projective spaces, Pacific J. Math. 113 (1984), 35–49.CrossRefGoogle Scholar
4.Hurwitz, A., Über der Komposition der quadratischer Formen, Math. Ann. 88 (1923), 1–25.CrossRefGoogle Scholar
5.James, I. and Thomas, E., An approach to the enumeration problem for non-stable vector bundles, J. Math. Mech. 14 (1965), 485–506.Google Scholar
6.Lam, K. Y. and Yiu, P. Y. H., Sums of squares formulae near the Hurwitz–Radon range, Contemp. Math. 58 (1987), 51–56.CrossRefGoogle Scholar
7.Looijenga, E., A note on polynomial isolated singularities, Nederl. Akad. Wetensch. Proc. A 74, Indag. Math. 33 (1971), 418–421.CrossRefGoogle Scholar
8.Milnor, J. W., Singular points of complex hypersurfaces (Princeton University Press, 1968).Google Scholar
9.Perron, B., Le noeud ‘huit’ est algebrique reel, Invent. Math. 65 (1981/1982), 441–451.CrossRefGoogle Scholar
10.Radon, J., Lineare scharen orthogonalen Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922), 1–14.CrossRefGoogle Scholar
11.Rees, E. G., Linear spaces of real matrices of given rank, Contemp. Math. 188 (1995), 219–229.CrossRefGoogle Scholar
12.Rees, E. G., Linear spaces of real matrices, in Proc. A. C. Aitken Centenary Conf., Dunedin, 1995; Otago Conf. Series 5 (ed. Kavalieris, L., Lam, F. C., Roberts, L. A. and Shanks, J. A.), pp. 311–320 (University of Otago Press, 1996).Google Scholar
13.Seade, A., Fibred links and a construction of real singularities via complex geometry, Bol. Soc. Brasil. Mat. (NS) 27 (1996), 199–215.CrossRefGoogle Scholar
14.Yoshida, T., On the vector bundles m ξn over real projective spaces, J. Sci. Hiroshima Univ. Ser. A-I Math. 32 (1968), 5–16.Google Scholar