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Cross-sections of the sphere and J-theory Mauro, David J.


Let F denote the field R or C. Let 0[sub n,k] denote the manifold of orthonormal k-frames in Fⁿ, where 1 ≤ k ≤ n. We can fibre over 0[sub n,k] by taking the last vector in each k-frame. Here d = 1 or 2, according to whether F equals R or C. A cross-section, S[sup dn-1] →0[sub n,k], assigns to each point u ε S[sup dn-1] an orthonormal k-frame (u₁, u₂, + +, u[sub k-1], u). We wish to determine values for n and k which will guarantee the existence of such a cross-section. In the real case this is the classical vector fields on spheres problem. Atiyah and James prove that the cross-sectioning problem is equivalent to a problem in J-theory. Let J (X) denote the set of equivalence classes of stable fibre homotopic orthogonal sphere bundles over a finte CW-complex X. It is well known that J (X) is a finite abelian group. If α is an F - vector bundle over X, we can associate with it a unique sphere bundle which we will denote by (α). The class of (α) will be denoted by J (α). Let ξ denote the canonical real d-dimensional Hopf line bundle over the F - projective space P[sup k]. The link between cross-sectioning and J-theory can be stated as follows: The Stiefel fibring 0[sub n,k] →S[sup dn-1] admits a cross-section if and only if n is a multiple of the order of J (ξ) in J (P[sup k]). Thus the problem of finding cross-sections has been reduced to determining the J-order of ξ, in J (P[sup k]).

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