Higher obstructions to sectioning a special type of fibre bundles

Abstract

ship of these secondary obstruction classes when m > 2n. In this paper, we shall study the higher obstructions to sectioning a special type of fibre bundles such that 72(F) is the first nontrivial homotopy group of the fibre and the dimension of the next homotopy group of F is much higher than 4. Let F be a homogeneous space of a compact connected semisimple Lie group G. Suppose that the first nontrivial homotopy group of F is m2(F) which is free. Let Q' be a bundle over X with fibre F and group G and whose primary obstruction vanishes. We will construct a principal torus bundle F, -- F, a central extension G1 of G by the torus, and a set of weakly associated bundles {4 over X with fibre F1 and group G1. Under the projection p: 4 -, secondary and tertiarv obstructions for 4 corresponding to primary and secondary obstructions for ('s (cf. Theorem (4.3)). We shall call this method enlarging the fibre and the group. Thus, an enlargement may be used to reduce the order of an obstruction. By this method, we rederive Kundert's formula for the secondary obstructions when the fibre is a complex projective space CP'-1 and PU(n) is the structural group(2). We will also derive new formulas for the third obstructions when F = CPI'- 1 and for the secondary obstructions mod 2 to finding an orientable 2plane sub-bundle of an orientable vector space bundle (cf. Theorems (6.1) and (7.1)). The author extends his cordial thanks to Professor N. E. Steenrod for his constant encouragement, suggestions and criticisms. He equally thanks Professors J. C. Moore, J. W. Milnor and Dr. P. F. Baum for many helpful discussions.