Microbundle

In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964.^[1] 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

The precise definition of a microbundle follows. Let B be a topological space. Then a n-microbundle consists of a triple (E, i, p) where E is a topological space (the "total space"), i is a map from B to E (the "zero section"), and p is a map from E to B ("the projection map"). Furthermore there are two conditions:

1. the composition of i followed by p must be the identity;
2. for every b in B, there must be a neighborhood V_b in E such that p restricted to V_b looks like a projection U_B × Rⁿ → U_b.

Note that the first condition suggests i is the zero section of a vector bundle, while the second is like 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. E could look very wild away from that neighborhood. Also, the maps gluing together locally trivial patches of the microbundle may only overlap the fibers.

Results

Two microbundles are isomorphic if they have neighborhoods of their zero sections which are homeomorphic by a map which make the necessary maps commute. Typical bundle operations such as induced bundles under pullback exist.

A theorem of Kister and Mazur states that there is a neighborhood of the zero section which is actually a fiber bundle with fiber Rⁿ and structure group Homeo(Rⁿ,0), the group of homeomorphisms of Rⁿ 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.

For a manifold M, a topological manifold, there is a microbundle given by the diagonal map M → M × M and projection to the first coordinate. Taking the fiber bundle contained in it gives the topological tangent bundle. Intuitively, this bundle is obtained by taking a system of small charts for M, letting each chart U have a fiber U 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 Kirby–Siebenmann work on smooth structures and PL structures on higher dimensional manifolds.

References

1. ^ Milnor, John Willard (1964), "Microbundles. I", Topology. An International Journal of Mathematics 3: 53–80, doi:10.1016/0040-9383(64)90005-9, MR0161346 

• Switzer, Robert M. (2002), Algebraic topology---homotopy and homology, Classics in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42750-6, MR1886843 , see Chapter 14.