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Two-plane sub-bundles of nonorientable real vector-bundles. (English) Zbl 0602.55009

Let \(\zeta\) be a nonorientable m-plane bundle over a CW complex \(\chi\) of dimension m or less. Given a 2-plane bundle \(\eta\) over \(\chi\), we wish to know whether \(\eta\) can be embedded as a sub-bundle of \(\zeta\). The bundle \(\eta\) need not be orientable. When \(\zeta\) is even-dimensional there is the added complication of twisted coefficients. In that case, we use Postnikov deomposition of certain nonsimple fibrations in order to describe the obstructions for the embedding problem. E. Thomas [Ann. Math., II. Ser. 86, 349-361 (1967; Zbl 0168.214); Invent. Math. 3, 334-347 (1967; Zbl 0162.554)] treated this problem for \(\zeta\) and \(\eta\) both orientable. The results are applied to the tangent bundle of a closed, connected, nonorientable smooth manifold, as a special case.

MSC:

55R25 Sphere bundles and vector bundles in algebraic topology
55S40 Sectioning fiber spaces and bundles in algebraic topology
55S45 Postnikov systems, \(k\)-invariants
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References:

[1] JAMES, I.M.: The topology of Stiefel manifolds. London Math. Soc. Lecture Note Series24, Cambridge Univ. Press (1976)
[2] KOSCHORKE, U.: Vector fields and other vector bundle morphims-a singularity approach. Lectures Notes in Math.847, Heidelberg: Springer-Verlag (1981) · Zbl 0459.57016
[3] MC CLENDON, J.F.: Obstruction theory in fiber spaces. Math. Z.120, 1-17 (1971) · Zbl 0222.55023 · doi:10.1007/BF01109713
[4] MELLO, M.H.P.L.: Doctoral Dissertation. Pontifícia Universidade Católica do Rio de Janeiro (1985)
[5] PAECHTER, G.F.: The groups ?r(Vn.m) (I). Quart. J. Math. Oxford (2)7, 249-268 (1956) · Zbl 0073.18402 · doi:10.1093/qmath/7.1.249
[6] POLLINA, B.J.: Tangent 2-fields on even-dimensional nonorientable manifolds. Trans. Amer. Math. Soc.271, 215-224 (1982) · Zbl 0492.57008
[7] RANDALL, D.: CAT 2-fields on nonorientable CAT manifolds (to appear) · Zbl 0628.57015
[8] ROBINSON, C.A.: Moore-Postnikov systems for non-simple fibrations. Illinois J. Math.16, 234-242 (1972) · Zbl 0227.55019
[9] THOMAS, E.: Seminar on fiber spaces. Lectures Notes in Math.13, Heidelberg: Springer-Verlag (1966)
[10] THOMAS, E.: Postnikov invariants and higher order cohomology operations. Ann. of Math.85, 184-217 (1967) · Zbl 0152.22002 · doi:10.2307/1970439
[11] THOMAS, E.: Fields of tangent 2-planes on even-dimensional manifolds. Ann. of Math.86, 349-361 (1967) · Zbl 0168.21401 · doi:10.2307/1970692
[12] THOMAS, E.: Fields of tangent k-planes on manifolds. Inventiones Math.3, 334-347 (1967) · Zbl 0162.55402 · doi:10.1007/BF01402957
[13] WHITEHEAD, G.W.: Elements of homotopy theory, 1st ed. Berlin-Heidelberg-New York: Springer-Verlag (1978) · Zbl 0406.55001
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