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The concept of a microbundle of dimension was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later Kister [Kister1964], and independently Mazur, showed that every microbundle uniquely determines a topological -bundle; i.e. a fibre bundle with structure group the homeomorphisms of fixing .
Definition 1.1 [Milnor1964] . Let be a topological space. An -dimensional microbundle over is a quadruple where is a space, and are maps fitting into the following diagram
and the following conditions hold:
- For all there exist open neigbourhood , an open neighbourhood of and a homeomorphism
which makes the following diagram commute:
The space is called the total space of the bundle and the base space.
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute:
An important example of a microbundle is the tangent microbundle of a topological (or similarly ) manifold . Let
be the diagonal map for .
Example 2.1 [Milnor1964, Lemma 2.1]. Let be topological (or PL) -manifold, and let be the projection onto the first factor. Then
is an -dimensional microbundle, the tangent microbundle of .
Remark 2.2. An atlas of gives a product atlas of which shows that the second condition of a microbundle is fulfilled. Actually the definition of the micro tangent bundle looks a bit more like a normal bundle to the diagonal, a view which fits to the fact that the normal bundle of the diagonal of a smooth manifold in is isomorphic to its tangent bundle.
Another important example of a microbundle is the micro-bundle defined by a topological topological -bundle.
Example 2.3. Let be a topological -bundle with zero section . Then the quadruple
is an -dimensional microbundle.
3 The Kister-Mazur Theorem
A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem, proven independently by Kister and Mazur.
Theorem 3.1 [Kister1964, Theorem 2]. Let be an -dimensional microbundle over a locally finite, finite dimensional simplicial complex . Then there is a neighbourhood of , such that the following hold.
- is the total space of a topological -bundle over .
- is a microbundle and the the inclusion is a microbundle isomorphism.
- If is any other such neighbourhood of then there is a -bundle isomorphism .
Remark 3.2. Microbundle theory is an important part of the work by Kirby and Siebenmann [Kirby&Siebenmann1977] on smooth structures and -structures on higher dimensional topological manifolds.
- [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004
- [Kister1964] J. M. Kister, Microbundles are fibre bundles, Ann. of Math. (2) 80 (1964), 190–199. MR0180986 (31 #5216) Zbl 0131.20602
- [Milnor1964] J. Milnor, Microbundles. I, Topology 3 (1964), no.suppl. 1, 53–80. MR0161346 (28 #4553b) Zbl 0124.38404
- The Wikipedia page about microbundles.