# Orientation of a vector bundle

In mathematics, an **orientation** of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: *E* →*B*, an orientation of *E* means: for each fiber *E*_{x}, there is an orientation of the vector space *E*_{x} and one demands that each trivialization map (which is a bundle map)

is fiberwise orientation-preserving, where **R**^{n} is given the standard orientation. In more concise terms, this says that the structure group of the frame bundle of *E*, which is the real general linear group *GL*_{n}(**R**), can be reduced to the subgroup consisting of those with positive determinant.

If *E* is a real vector bundle of rank *n*, then a choice of metric on *E* amounts to a reduction of the structure group to the orthogonal group *O*(*n*). In that situation, an orientation of *E* amounts to a reduction from *O*(*n*) to the special orthogonal group *SO*(*n*).

A vector bundle together with an orientation is called an **oriented bundle**. A vector bundle that can be given an orientation is called an **orientable vector bundle**.

The basic invariant of an oriented bundle is the Euler class. The multiplication (that is, cup product) by the Euler class of an oriented bundle gives rise to a Gysin sequence.

## Examples

A complex vector bundle is oriented in a canonical way.

The notion of an orientation of a vector bundle generalizes an orientation of a manifold: an orientation of a manifold is an orientation of the tangent bundle of the manifold. In particular, a manifold is orientable if and only if its tangent bundle is orientable as a vector bundle. (note: as a manifold, a tangent bundle is always orientable.)

## Operations

To give an orientation to a real vector bundle *E* of rank *n* is to give an orientation to the (real) determinant bundle of *E*. Similarly, to give an orientation to *E* is to give an orientation to the unit sphere bundle of *E*.

Just as a real vector bundle is classified by the real infinite Grassmannian, oriented bundles are classified by the infinite Grassmannian of oriented real vector spaces.

## Thom space

From the cohomological point of view, for any ring Λ, a Λ-orientation of a real vector bundle *E* of rank *n* means a choice (and existence) of a class

in the cohomology ring of the Thom space *T*(*E*) such that *u* generates as a free -module globally and locally: i.e.,

is an isomorphism (called the Thom isomorphism), where "tilde" means reduced cohomology, that restricts to each isomorphism

induced by the trivialization . One can show, with some work, that the usual notion of an orientation coincides with a **Z**-orientation.

## See also

- The integration along the fiber
- Orientation bundle (or orientation sheaf) - this is used to formulate the Thom isomorphism for non-oriented bundles.

## References

- Bott, Raoul; Tu, Loring (1982),
*Differential Forms in Algebraic Topology*, New York: Springer, ISBN 0-387-90613-4 - J.P. May,
*A Concise Course in Algebraic Topology.*University of Chicago Press, 1999. - Milnor, John Willard; Stasheff, James D. (1974),
*Characteristic classes*, Annals of Mathematics Studies,**76**, Princeton University Press; University of Tokyo Press, ISBN 978-0-691-08122-9