Vector Bundles and their Related Algebra
by
M. R. Adhikari
The University of Burdwan

The notion of vector bundles arose from the study of tangent vector fields to smooth geometric objects e.g., spheres, projective spaces, and more generally manifolds.

Let X be a topological space and Vect_F(X) be the set of isomorphism classes of F-Vector bundles over X where F = R or C or H. Then Vect_F(X) admits a commutative semigroup structure under addition function (\alpha, \beta) --> \alpha\oplus\beta, the Whitney sum of two Vector bundles \alpha and \beta over X (using the same symbol \alpha for a Vector bundle and also for its isomorphism class). Then Vect_F is a contravariant functor from the category of topological spaces and continuous maps to the category s of abelian semigroups and homomorphisms. It is also a contravariant functor from the homotopy category of paracompact spaces to s. By using Yoneda's Lemma a representation of Vect_F and a classification of Vect_F(X) are obtained.

For F = R or C, Vect_F(X) endowed with compositions \oplus and \otimes forms a semiring where \otimes denotes the tensor product of two Vector bundles over X. Then it has a unique ring completion K_F(X). Finally, we study the ideals of Vect_F(X) and K_F(X) and relate them.

1 M. R. Adhikari Algebra and Applications (in press)

2 M. R. Adhikari An application of Yoneda's lemma in semirings Bull. Cal. Math. Sci. 86 1994 15-20

3 M. R. Adhikari and M. K. Das A classification of vector bundles News Bull. Cal. Math. Soc. 15(8) 1993 10-11

4 M. R. Adhikari and P. Das Ideals associated with Vector Bundles (communicated)

5 D. Husemoller Fibre bundles Springer Verlag 1966

6 M. K. Sen and M. R. Adhikari On maximal ideals of semirings Proc. Amer. Math. Soc. 118 1993 699-703

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caah-02.