Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-01T00:09:59.529Z Has data issue: false hasContentIssue false
  • English
  • Français

On the KOℤ/2-Euler class, I

Published online by Cambridge University Press:  14 November 2011

M. C. Crabb
M. C. Crabb
Affiliation:
Department of Mathematics, University of Aberdeen, Aberdeen AB9 2TY, Scotland, U.K
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper presents a systematic account of ℤ/2-equivariant KO-theoretic methods in the study of r-fields (that is, r linearly independent cross-sections, r ≧ 1) on a real vector bundle. Applications of the theory to the problem of immersing complex and quaternionic projective spaces in Euclidean space are given in [13].

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 1-2 , 1991 , pp. 115 - 137
Copyright
Copyright © Royal Society of Edinburgh 1991

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

1Adams, J. F.. Vector fields on spheres. Ann. of Math. 75 (1962), 603–632.CrossRefGoogle Scholar
2Adams, J. F.. On the groups J(X), IV. Topology 5 (1966), 21–71.Google Scholar
3Adams, J. F.. Stable homotopy and generalised homology (Chicago: University of Chicago Press, 1974).Google Scholar
4Atiyah, M. F.. Power operations in K-theory. Quart. J. Math. Oxford (2) 17 (1966), 165–193.CrossRefGoogle Scholar
5Atiyah, M. F.. K-theory (New York: Benjamin, 1967).Google Scholar
6Atiyah, M. F.. Vector fields on manifolds (Düsseldorf: Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 200, 1970).CrossRefGoogle Scholar
7Atiyah, M. F.. and Dupont, J. L.. Vector fields with finite singularities. Acta Math. 128 (1972), 1–40.CrossRefGoogle Scholar
8Atiyah, M. F.. and Hirzebruch, F.. Quelques thé'rèmes de non-plongement pour les varié'té's différentiables. Bull. Soc. Math. France 87 (1959), 383–396.Google Scholar
9Becker, J. C.. Cohomology and the classification of liftings. Trans. Amer. Math. Soc. 133 (1968), 447–475.Google Scholar
10Crabb, M. C.. The Euler class, the Eider characteristic and obstruction theory for monomorphisms of vector bundles (D. Phil. Thesis, University of Oxford, 1975).Google Scholar
11Crabb, M. C.. ℤ/2-Homotopy Theory (Edinburgh: Cambridge University Press, 1980).Google Scholar
12Crabb, M. C.. On the KOℤ/2-Euler class, II. Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), 139–154.CrossRefGoogle Scholar
13Crabb, M. C.. Immersing projective spaces in Euclidean space Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), 155–170.CrossRefGoogle Scholar
14Crabb, M. C.. and Knapp, K.. On the codegree of negative multiples of the Hopf bundle. Proc. Roy. Soc. Edinburgh Sect. A 107 (1987), 87–107.Google Scholar
15Crabb, M. C.. and Steer, B.. Vector-bundle monomorphisms with finite singularities. Proc. London Math. Soc. 30 (1975), 1–39.CrossRefGoogle Scholar
16Crabb, M. C.. and Sutherland, W. A.. The space of sections of a sphere-bundle, I. Proc. Edinburgh Math. Soc. 29 (1986), 383–403.Google Scholar
17Davis, D. M.. Generalised homology and the generalised vector field problem. Quart. J. Math. Oxford 25 (1974), 169–194.Google Scholar
18Davis, D. M.. and Mahowald, M.. The Euler class for connective KO-theory and an application to immersions of quaternionic projective space. Indiana Univ. Math. J. 28 (1979), 1025–1034.Google Scholar
19Dax, J. P.. Etude homotopique des espaces de plongements. Ann. Sci. Ecole Norm. Sup. (4) 5 (1972), 303–377.Google Scholar
20Dieck, T. torn. Transformation Groups and Representation Theory. Lecture Notes in Mathematics 766 (Berlin: Springer, 1979).Google Scholar
21Feder, S.. Immersions and embeddings in complex projective spaces. Topology 4 (1965), 143–158.CrossRefGoogle Scholar
22Feder, S.. and Iberkleid, W.. Secondary operations in K-theory and the generalized vector field problem. Lecture Notes in Math. 597 (1977), 161–175.CrossRefGoogle Scholar
23Haefliger, A.. and Hirsch, M. W.. Immersions in the stable range. Ann. of Math. 75 (1962), 231–241.CrossRefGoogle Scholar
24James, I. M.. Spaces associated with Stiefel manifolds. Proc. London Math. Soc. 9 (1959), 115–140.CrossRefGoogle Scholar
25James, I. M.. On the immersion problem for real projective spaces. Bull. Amer. Math. Soc. 69 (1963), 231–238.CrossRefGoogle Scholar
26Koschorke, U.. Framefields and nondegenerate singularities. Bull. Amer. Math. Soc. 81 (1975), 157–160.CrossRefGoogle Scholar
27Larmore, L. L.. Twisted cohomology theories and the single obstruction to lifting. Pacific J. Math. 41 (1972), 755–769.Google Scholar
28Mahowald, M. E.. The metastable homotopy of S n. American Mathematical Society Memoir 72 (Providence, R. I.: American Mathematical Society, 1967).Google Scholar
29Mayer, K. H.. Elliptische Differentialoperatoren und Ganzzahligkeitssätze für charakteristische Zahlen. Topology 4 (1965), 295–313.CrossRefGoogle Scholar
30Salomonsen, H. A.. Bordism and geometric dimension. Math. Scand. 32 (1973), 87–111.Google Scholar
31Segal, G. B.. Equivariant K-theory. Inst. Hautes Etudes Sci. Publ. Math. 34 (1968), 129–151.CrossRefGoogle Scholar
32Sigrist, F.. and Suter, U.. On immersions CP n ↛R4n-2α(n). Lecture Notes in Math. 673 (1978), 106–115.Google Scholar
33Steer, B.. Une interprétation géométrique des nombres de Radon-Hurwitz. Ann. Inst. Fourier (Grenoble) 17 (1967), 209–218.CrossRefGoogle Scholar
34Steer, B.. On immersing complex projective (4k + 3)-space in euclidean space. Quart. J. Math. Oxford 22 (1971), 339–345.Google Scholar
35Stong, R. E.. Notes on Cobordism Theory (Princeton: Princeton University Press, 1968).Google Scholar