Microbundle - Results

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.