Microbundle
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1 Introduction
The concept of 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.
Definition 1.1 [Milnor1964] .
An -dimensional microbundle is a quadrupleTex syntax errorsuch that there is a sequence
Tex syntax error
Tex syntax error
- for all
Tex syntax error
there exist open neigbourhoodTex syntax error
, an open neighbourhoodTex syntax error
ofTex syntax error
and a homeomorphismTex syntax error
which makes the following diagram commute:
Example 1.2 [Milnor1964, Lemma 2.1].
Let be topological -manifold, let be the diagonal map and letTex syntax errorbe the projection onto the first factor. Then
Tex syntax error
is an -dimensional microbundle.
Example 1.3. Let
Tex syntax errorbe a topological -bundle with zero section
Tex syntax error. Then
Tex syntax error
is an -dimensional microbundle.
Definition 1.4.
Two microbundlesTex syntax error, over the same space are isomorphic if there exist neighbourhoods
Tex syntax errorof
Tex syntax errorand
Tex syntax errorof
Tex syntax errorand a homeomorphism
Tex syntax errormaking the following diagram commute.
Tex syntax error
Theorem 1.5 [Kister1964, Theorem 2] .
LetTex syntax errorbe an -dimensional microbundle. Then there is a neighbourhood of
Tex syntax error,
Tex syntax errorsuch that:
- is the total space of a topological -bundle over .
- The inclusion
Tex syntax error
is a microbundle isomorphism - If
Tex syntax error
is any other such neighbourhood ofTex syntax error
then there is a -bundle isomorphismTex syntax error
.
2 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