In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964. It allows the creation of bundle-like objects in situations where they would not ordinarily be thought to exist. For example, the tangent bundle is defined for a smooth manifold but not a topological manifold; use of microbundles allows the definition of a topological tangent bundle.

Definition

A (topological) -microbundle over a topological space , where is a topological space (the "total space"), and are continuous maps (respectively, the "zero section" and the "projection map") such that:

1. the composition is the identity of ;

2. for every , there are a neighborhood is homeomorphic and the maps and commute with and .

In analogy with vector bundles, the integer is also called the rank or the fibre dimension of the microbundle. Similarly, note that the first condition suggests should be thought of as the zero section of a vector bundle, while the second mimics the local triviality condition on a bundle. An important distinction here is that "local triviality" for microbundles only holds near a neighborhood of the zero section. The space could look very wild away from that neighborhood. Also, the maps gluing together locally trivial patches of the microbundle may only overlap the fibers.

The definition of microbundle can be adapted to other categories more general than the smooth one, such as that of piecewise linear manifolds, by replacing topological spaces and continuous maps by suitable objects and morphisms.

Examples

• Any vector bundle has an obvious -microbundle is the zero section.

• Given any topological space , the cartesian product (together with the projection on and the map ) defines an -microbundle, called the standard trivial microbundle.

• Given a topological manifold together with the projection on the first component and the diagonal map defines an -microbundle, called the tangent microbundle.

• Given an over defines an , called the pullback (or induced) microbundle, together with the projection and the zero section is a vector bundle, the pullback microbundle of its underlying microbundle is precisely the underlying microbundle of the standard pullback bundle.

• Given an over , the restricted microbundle.

Morphisms

are isomorphic, together with a homeomorphism commuting with the projections and the zero sections.

More generally, a morphism between microbundles consists of a germ between neighbourhoods of the zero sections as above.

-microbundle is called trivial. The local triviality condition in the definition of microbundle can therefore be restated as follows: for every there is a neighbourhood is trivial.

Analogously to parallelisable smooth manifolds, a topological manifold is called topologically parallelisable if its tangent microbundle is trivial.

Properties

A theorem of James Kister and Barry Mazur states that there is a neighborhood of the zero section which is actually a fiber bundle and structure group fixing the origin. This neighborhood is unique up to isotopy. Thus every microbundle can be refined to an actual fiber bundle in an essentially unique way.

gives the topological tangent bundle. Intuitively, this bundle is obtained by taking a system of small charts for , letting each chart have a fiber over each point in the chart, and gluing these trivial bundles together by overlapping the fibers according to the transition maps.

Microbundle theory is an integral part of the work of Robion Kirby and Laurent C. Siebenmann on smooth structures and PL structures on higher dimensional manifolds.

References

• See Chapter 14.

External links

• Microbundle at the Manifold Atlas.