Abstract
Let ζ be a nonorientable m-plane bundle over a CW complex X of dimension m or less. Given a 2-plane bundle η over X, we wish to know whether η can be embedded as a sub-bundle of ζ. The bundle η need not be orientable. When ζ is even-dimensional there is the added complication of twisted coefficients. In that case, we use Postnikov decomposition of certain nonsimple fibrations in order to describe the obstructions for the embedding problem. Emery Thomas [11] and [12] treated this problem for ζ and η both orientable. The results found here are applied to the tangent bundle of a closed, connected, nonorientable smooth manifold, as a special case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
JAMES, I.M.: The topology of Stiefel manifolds. London Math. Soc. Lecture Note Series24, Cambridge Univ. Press (1976)
KOSCHORKE, U.: Vector fields and other vector bundle morphims-a singularity approach. Lectures Notes in Math.847, Heidelberg: Springer-Verlag (1981)
MC CLENDON, J.F.: Obstruction theory in fiber spaces. Math. Z.120, 1–17 (1971)
MELLO, M.H.P.L.: Doctoral Dissertation. Pontifícia Universidade Católica do Rio de Janeiro (1985)
PAECHTER, G.F.: The groups πr(Vn.m) (I). Quart. J. Math. Oxford (2)7, 249–268 (1956)
POLLINA, B.J.: Tangent 2-fields on even-dimensional nonorientable manifolds. Trans. Amer. Math. Soc.271, 215–224 (1982)
RANDALL, D.: CAT 2-fields on nonorientable CAT manifolds (to appear)
ROBINSON, C.A.: Moore-Postnikov systems for non-simple fibrations. Illinois J. Math.16, 234–242 (1972)
THOMAS, E.: Seminar on fiber spaces. Lectures Notes in Math.13, Heidelberg: Springer-Verlag (1966)
THOMAS, E.: Postnikov invariants and higher order cohomology operations. Ann. of Math.85, 184–217 (1967)
THOMAS, E.: Fields of tangent 2-planes on even-dimensional manifolds. Ann. of Math.86, 349–361 (1967)
THOMAS, E.: Fields of tangent k-planes on manifolds. Inventiones Math.3, 334–347 (1967)
WHITEHEAD, G.W.: Elements of homotopy theory, 1st ed. Berlin-Heidelberg-New York: Springer-Verlag (1978)
Author information
Authors and Affiliations
Additional information
The writing of this paper was partially supported by CNPq grant
Rights and permissions
About this article
Cite this article
de Paula Leite Mello, M.H. Two-plane sub-bundles of nonorientable real vector-bundles. Manuscripta Math 57, 263–280 (1987). https://doi.org/10.1007/BF01437484
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01437484