Microbundle
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For any space $X$ define the diagonal embedding | For any space $X$ define the diagonal embedding | ||
$$\Delta_X \colon X \to X \times X, \quad x \mapsto (x,x)~.$$ | $$\Delta_X \colon X \to X \times X, \quad x \mapsto (x,x)~.$$ |
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1 Definition
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 [Kister1964] 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:
2 Examples
For any space define the diagonal embedding
If is a differentiable -manifold the normal bundle of is the tangent bundle of . In the topological category we have the following definition.
Example 2.1 [Milnor1964, Lemma 2.1]. Let be topological -manifold, and let be the projection onto the first factor. Then
is an -dimensional microbundle, the tangent microbundle of .
is an -dimensional microbundle.
Theorem 2.3 [Kister1964, Theorem 2]. Let be an -dimensional microbundle. Then there is a neighbourhood of , such that:
- is the total space of a topological -bundle over .
- The inclusion is a microbundle isomorphism
- If is any other such neighbourhood of then there is a -bundle isomorphism .
3 References
- [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
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:
2 Examples
For any space define the diagonal embedding
If is a differentiable -manifold the normal bundle of is the tangent bundle of . In the topological category we have the following definition.
Example 2.1 [Milnor1964, Lemma 2.1]. Let be topological -manifold, and let be the projection onto the first factor. Then
is an -dimensional microbundle, the tangent microbundle of .
is an -dimensional microbundle.
Theorem 2.3 [Kister1964, Theorem 2]. Let be an -dimensional microbundle. Then there is a neighbourhood of , such that:
- is the total space of a topological -bundle over .
- The inclusion is a microbundle isomorphism
- If is any other such neighbourhood of then there is a -bundle isomorphism .
3 References
- [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