 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 Crosssections of the sphere and Jtheory
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
Crosssections of the sphere and Jtheory Mauro, David J.
Abstract
Let F denote the field R or C. Let 0[sub n,k] denote the manifold of orthonormal kframes in Fⁿ, where 1 ≤ k ≤ n. We can fibre over 0[sub n,k] by taking the last vector in each kframe. Here d = 1 or 2, according to whether F equals R or C. A crosssection, S[sup dn1] →0[sub n,k], assigns to each point u ε S[sup dn1] an orthonormal kframe (u₁, u₂, + +, u[sub k1], u). We wish to determine values for n and k which will guarantee the existence of such a crosssection. In the real case this is the classical vector fields on spheres problem. Atiyah and James prove that the crosssectioning problem is equivalent to a problem in Jtheory. Let J (X) denote the set of equivalence classes of stable fibre homotopic orthogonal sphere bundles over a finte CWcomplex 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 ddimensional Hopf line bundle over the F  projective space P[sup k]. The link between crosssectioning and Jtheory can be stated as follows: The Stiefel fibring 0[sub n,k] →S[sup dn1] admits a crosssection if and only if n is a multiple of the order of J (ξ) in J (P[sup k]). Thus the problem of finding crosssections has been reduced to determining the Jorder of ξ, in J (P[sup k]).
Item Metadata
Title 
Crosssections of the sphere and Jtheory

Creator  
Publisher 
University of British Columbia

Date Issued 
1981

Description 
Let F denote the field R or C. Let 0[sub n,k] denote the manifold of orthonormal kframes in Fⁿ, where 1 ≤ k ≤ n. We can fibre over 0[sub n,k] by taking the last vector in each kframe. Here d = 1 or 2, according to whether F equals R or C. A crosssection, S[sup dn1] →0[sub n,k], assigns to each point u ε S[sup dn1] an orthonormal kframe (u₁, u₂, + +, u[sub k1], u). We wish to determine values for n and k which will guarantee the existence of such a crosssection. In the real case this is the classical vector fields on spheres problem. Atiyah and James prove that the crosssectioning problem is equivalent to a problem in Jtheory. Let J (X) denote the set of equivalence classes of stable fibre homotopic orthogonal sphere bundles over a finte CWcomplex 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 ddimensional Hopf line bundle over the F  projective space P[sup k]. The link between crosssectioning and Jtheory can be stated as follows: The Stiefel fibring 0[sub n,k] →S[sup dn1] admits a crosssection if and only if n is a multiple of the order of J (ξ) in J (P[sup k]). Thus the problem of finding crosssections has been reduced to determining the Jorder of ξ, in J (P[sup k]).

Genre  
Type  
Language 
eng

Date Available 
20100326

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 
10.14288/1.0080169

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.