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. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/20514 |
| Date | January 1977 |
| Creators | Allard, Jacques |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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