Structure of finite dimensional linear bundles morphisms (original) (raw)
A note on nonstable monomorphisms¶of vector bundles
manuscripta mathematica, 2000
In this paper we consider the question of the existence of a nonstable vector bundle monomorphism u : α → β over a closed, connected and smooth manifold M, when dimension of α = 3, dimension of β = dimension of M = n ≡ 0(4). The singularity method provides the full obstruction to this problem and under some homological hypothesis we can compute it in terms of well known invariants. an auxiliary virtual vector bundle over M. We can assume (because of the dimension hypothesis) that there is a generic vector bundle homomorphism u : α → β which has rank at least two in each point of M. The singularity is the set S := { x ∈ M | rank(u x : α x → β x) = 2}.
Some results on vector bundle monomorphisms
Algebraic & Geometric Topology, 2007
In this paper we use the singularity method of Koschorke [2] to study the question of how many different nonstable homotopy classes of monomorphisms of vector bundles lie in a stable class and the percentage of stable monomorphisms which are not homotopic to stabilized nonstable monomorphisms. Particular attention is paid to tangent vector fields. This work complements some results of Koschorke [3; 4], Libardi-Rossini [7] and Libardi-do Nascimento-Rossini [6].
Vector bundles and regulous maps
Mathematische Zeitschrift, 2013
Let X be a compact nonsingular affine real algebraic variety. We prove that every pre-algebraic vector bundle on X becomes algebraic after finitely many blowing ups. Using this theorem, we then prove that the Stiefel-Whitney classes of any pre-algebraic R-vector bundle on X are algebraic. We also derive that the Chern classes of any pre-algebraic C-vector bundles and the Pontryagin classes of any pre-algebraic R-vector bundle are blow-C-algebraic. We also provide several results on line bundles on X .
Lecture Notes in Physics, 2008
The notion of vector bundle is a basic extension to the geometric domain of the fundamental idea of a vector space. Given a space X, we take a real or complex finite dimensional vector space V and make V the fibre of a bundle over X, where each fibre is isomorphic to this vector space. The simplest way to do this is to form the product X × V and the projection pr X : X × V → X onto the first factor. This is the product vector bundle with base X and fibre V .
On a Lie Algebraic Characterization of Vector Bundles
Symmetry, Integrability and Geometry: Methods and Applications, 2012
We prove that a vector bundle π : E → M is characterized by the Lie algebra generated by all differential operators on E which are eigenvectors of the Lie derivative in the direction of the Euler vector field. Our result is of Pursell-Shanks type but it is remarkable in the sense that it is the whole fibration that is characterized here. The proof relies on a theorem of [Lecomte P., J. Math. Pures Appl. (9) 60 (1981), 229-239] and inherits the same hypotheses. In particular, our characterization holds only for vector bundles of rank greater than 1.
The Topology of Fiber Bundles Lecture Notes
1998
5.4. Applications II: H * (ΩS n) and H * (U (n)) 162 5.5. Applications III: Spin and Spin C structures 165 Bibliography v Corollary 1.9. A vector bundle ζ : p : E → B is trivial if and only if its associated principal GL(n)-bundle p : E GL → B admits a section.