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Contribution to the theory of stably trivial vector bundles Allard, Jacques


A vector bundle E over a CW-complex X is said to be stably trivial of type (n,k) if E, © ke = ne, where e denotes the trivial line bundle. Let V[sub n,k] , be the Stiefel manifold of orthonormal k- frame in euclidian n-space R[sup n] and let n[sub n,k] , be the real (n-k)- dimensional vector bundle over V[sub n,l] , whose fiber over a k—frame x is the subspace of R[sup n] orthogonal to the span of the vectors in x . The vector bundle n[sub n,k], is "weakly universal" for stably trivial vector bundles of type (n,k), i.e. for any stably trivial vector bundle of type (n,k), there is a map f: X -> V[sub n,k] , not necessarily unique up to homotopy, such that f *n[sub n,k] = E, . We study the following questions: (a) for which values of r is the r-fold Whitney sum rn[sub n,k] , trivial, and (b) what is the maximum number of linearly independent cross-sections of n[sub n,k] ©[sup se] (0 < s < k - 1) . Among the results obtained are: (1) 2n[sub n,2] is trivial iff n is even or n=3; (2) 3n[sub n,2] is trivial if n is even; (3) rn[sub n,k], is not trivial if r is odd and < (n-2)/(n-k): (4) n[sub n,k] © (k-l)c is not trivial if n ≄ 2,4,8 and 1 < k < n - 3; (5) n[sub n,k] © [sup se] admits exactly s linearly independent cross-sections if n and k are odd; (6) n[sub n,k] © (k-2)e admits at most (k-1) linearly independent sections if 2<_k<_n-3. These results are used to construct examples of stably free modules and unimodular matrices over commutative noetherian rings. The techniques used are those of homotopy theory, including Postnikov systems, K-theory and, specially, Spin operations on vector bundles. A chapter of the thesis is devoted to defining the Spin operations formally as a type of K-theoretic characteristic classes for a certain type of real vector bundles. Formulae to compute the Spin operations on a Whitney sum of vector bundles are given.

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