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Contribution to the theory of stably trivial vector bundles Allard, Jacques
Abstract
A vector bundle E over a CWcomplex 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 nspace R[sup n] and let n[sub n,k] , be the real (nk) 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 rfold Whitney sum rn[sub n,k] , trivial, and (b) what is the maximum number of linearly independent crosssections 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 < (n2)/(nk): (4) n[sub n,k] © (kl)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 crosssections if n and k are odd; (6) n[sub n,k] © (k2)e admits at most (k1) linearly independent sections if 2<_k<_n3. 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, Ktheory and, specially, Spin operations on vector bundles. A chapter of the thesis is devoted to defining the Spin operations formally as a type of Ktheoretic 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.
Item Metadata
Title 
Contribution to the theory of stably trivial vector bundles

Creator  
Publisher 
University of British Columbia

Date Issued 
1977

Description 
A vector bundle E over a CWcomplex 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 nspace R[sup n] and let n[sub n,k] , be the real (nk)
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 rfold Whitney sum rn[sub n,k] , trivial, and (b) what is the maximum
number of linearly independent crosssections 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 < (n2)/(nk):
(4) n[sub n,k] © (kl)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 crosssections
if n and k are odd; (6) n[sub n,k] © (k2)e admits at most (k1)
linearly independent sections if 2<_k<_n3.
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, Ktheory and, specially, Spin operations on vector
bundles. A chapter of the thesis is devoted to defining the Spin operations formally as a type of Ktheoretic 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.

Genre  
Type  
Language 
eng

Date Available 
20100219

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080124

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.