The Topology of Quasibundles

Movahedi-Lankarani, H. ; Wells, R.

Canadian journal of mathematics, Tome 47 (1995) no. 6, pp. 1290-1316

Résumé

Let M(N, N) be the space of all N × N real matrices and let G(N) be the set of all linear subspaces of RN. The maps ker and coker from M(N, N) onto G(N) induce two quotient topologies, the right and left respectively. A quasibundle over a space X is defined as a continuous map from X into G(N)\ it is a right quasibundle if G(N) = M(N,N)/ ker and a left quasibundle if G(N) = M(N,N)/ coker. The following is established. Theorem: Let ξ be a left quasibundle over a closed subset of some Euclidean space. Then the following statements are equivalent: (i) ξ has enough sections pointwise. (ii) Sections zero at infinity over closed subsets may be extended globally, (iii) A vector subbundle over a closed subset extends to a vector subbundle over a neighborhood, (iv) ξ is a fibrewise sum of local vector subbundles. (v) There exist finitely many global sections spanning ξ. (vi) ξ is an image quasibundle. (vii) ξ results from a Swan construction. These results are used to prove a version of the Hirsch-Smale immersion theorem for locally compact subsets of Euclidean space.

Détail

DOI : 10.4153/CJM-1995-066-8

Mots-clés : Primary: 55R55, 55R65, 54C20, quasibundles, image bundles, kernel bundles, fibre, classifying space, monotopy, immersion

Bibliographie

[1] 1. Dugundji, J., Topology, Allyn and Bacon, Boston, 1966. Google Scholar

[2] 2. Fédérer, H., Geometric Measure Theory, Springer-Verlag, New York, 1969. Google Scholar

[3] 3. Glaeser, G., Étude de quelque algèbres tayloriennes, J. Analyse Math. 6(1958), 1–124.erratum, insert to (2) 6(1958). Google Scholar

[4] 4. Goresky, M. and MacPherson, R., Stratified Morse Theory, Springer-Verlag, New York 1988. Google Scholar

[5] 5. Gromov, M., Partial Differential Relations, Springer-Verlag, New York, 1986. Google Scholar

[6] 6. Helgason, S., Differential Geometry and Symmetric Spaces, Academic Press, New York, 1964. Google Scholar

[7] 7. Henon, M., A two dimensional mapping with a strange attractor, Comm. Math. Phys. 50(1976), 69–77. Google Scholar

[8] 8. Hirsch, M., Immersions of manifolds, Trans. Amer. Math. Soc. 93(1959), 242–276. Google Scholar

[9] 9. Hurewicz, W. and Wallman, H., Dimension Theory, Princeton University Press, 1941. Google Scholar

[10] 10. Husemoller, D., Fibre Bundles, McGraw-Hill, 1966. Google Scholar

[11] 11. Mansfield, R., Movahedi-Lankarani, H. and Wells, R., Smooth finite dimensional embeddings, preprint. Google Scholar

[12] 12. Morgan, F., Geometric Measure Theory—A Beginners Approach, Academic Press, San Diego, 1988. Google Scholar

[13] 13. Phillips, A., Submersions of open manifolds, Topology (2) 6(1967), 170–206. Google Scholar

[14] 14. Phillips, A., Publications I.R.M.A.—U.S.T.L.Flandres Artois, 1987. Volume 6, Numéro Spécial, Journées M.-H. Schwartz. Google Scholar

[15] 15. Samsonowicz, J., Images of vector bundles morphism, Bull. Polish Acad. Sci. Math 34(1986), 599–607. Google Scholar

[16] 16. Schwartz, M.H., Classes et caractères de Chern-Mather des espaces linéaires, C.R. Acad. Sci. Paris Sér. I Math. (5) 295(1982), 399–402. Google Scholar

[17] 17. Swan, R.G., Vector bundles and projective modules, Trans. Amer. Math. Soc. 105(1962), 264–277. Google Scholar