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Keywords:
CW-complex; oriented vector bundle; characteristic classes; Postnikov tower
CW-complex; oriented vector bundle; characteristic classes; Postnikov tower
Summary:
Necessary and sufficient conditions for the existence of $n$-dimensional oriented vector bundles ($n=3,4,5$) over CW-complexes of dimension $\le 7$ with prescribed Stiefel-Whitney classes $w_2=0$, $w_4 $ and Pontrjagin class $p_1$ are found. As a consequence some results on the span of 6 and 7-dimensional oriented vector bundles are given in terms of characteristic classes.
References:
[1] Čadek M., Vanžura J.: On the classification of oriented vector bundles over 5-complexes. preprint. MR 1258434
[2] Dold A., Whitney H.: Classification of oriented sphere bundles over a 4-complex. Ann. of Math. (1959), 69 667-677. MR 0123331 | Zbl 0124.38103
[3] James I., Thomas E.: An approach to the enumeration problem for non-stable vector bundles. J. Math. Mech. (1965), 14 485-506. MR 0175134 | Zbl 0142.40701
[4] Milnor J.: Some consequences of a theorem of Bott. Ann. of Math. (1958), 68 444-449. MR 0102805 | Zbl 0085.17301
[5] Tze-Beng Ng: On the geometric dimension of vector bundles, span of manifold and immersion of manifolds in manifolds. Exposition. Math. (1990), 8 193-226. MR 1062767
[6] Quillen D.: The mod 2 cohomology rings of extra-special 2-groups and their spinor groups. Math. Ann. (1971), 194 197-212. MR 0290401
[7] Thomas E.: On the cohomology of the real Grassmann complexes. Trans. Amer. Math. Soc. (1960), 96 67-89. MR 0121800 | Zbl 0098.36301
[8] Thomas E.: Homotopy classification of maps by cohomology homomorphisms. Trans. Amer. Math. Soc. (1964), 111 138-151. MR 0160212 | Zbl 0119.18401
[9] Thomas E.: Seminar on fibre bundles. Lecture Notes in Math., no 13 Springer Berlin - Heidelberg - New York (1966). MR 0203733
[10] Thomas E.: Postnikov invariants and higher order cohomology operation. Ann. of Math. (1967), 85 184-217. MR 0210135
[11] Thomas E.: Real and complex vector fields on manifolds. J. Math. Mech. (1967), 16 1183-1206. MR 0210136 | Zbl 0153.53503
[12] Thomas E.: Fields of $k$-planes on manifolds. Invent. Math. (1967), 3 334-347. MR 0217814
[13] Thomas E.: Vector fields on low dimensional manifolds. Math. Z. (1968), 103 85-93. MR 0224109 | Zbl 0162.55403
[14] Woodward L.M.: The classification of orientable vector bundles over CW-complexes of small dimension. Proc. Roy. Soc. Edinburgh (1982), 92A 175-179. MR 0677482 | Zbl 0505.55017
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