Microbundle
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== Introduction == | == Introduction == | ||
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The concept of a <i>microbundle</i> of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[topological manifold]]. Later \cite{Kister1964} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle. | The concept of a <i>microbundle</i> of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[topological manifold]]. Later \cite{Kister1964} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle. | ||
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Two microbundles $(E_n,B,i_n,j_n)$, $n=1,2$ over the same space $B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute. | Two microbundles $(E_n,B,i_n,j_n)$, $n=1,2$ over the same space $B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute. | ||
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Revision as of 12:21, 6 December 2012
An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 12:20, 16 May 2013 and the changes since publication. |
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1 Introduction
Tex syntax errorwas 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
Tex syntax error-bundle.
Definition 1.1 [Milnor1964] .
AnTex syntax error-dimensional microbundle is a quadruple
Tex 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:
Tex syntax error
Tex syntax errordefine the diagonal embedding
Tex syntax error
Tex syntax erroris a differentiable
Tex syntax error-manifold the normal bundle of
Tex syntax erroris the tangent bundle
Tex syntax errorof
Tex syntax error.
In the topological category we have:
Example 1.2 [Milnor1964, Lemma 2.1].
LetTex syntax errorbe topological
Tex syntax error-manifold, and let
Tex syntax errorbe the projection onto the first factor. Then
Tex syntax error
Tex syntax error-dimensional microbundle, the tangent microbundle
Tex syntax errorof
Tex syntax error.
Example 1.3. Let
is an
Tex syntax errorbe a topological
Tex syntax error-bundle with zero section
Tex syntax error. Then
Tex syntax error
Tex syntax error-dimensional microbundle.
Definition 1.4.
Two microbundlesTex syntax error,
Tex syntax errorover the same space
Tex syntax errorare 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
Tex syntax error-dimensional microbundle. Then there is a neighbourhood of
Tex syntax error,
Tex syntax errorsuch that:
-
Tex syntax error
is the total space of a topologicalTex syntax error
-bundle overTex syntax error
. - The inclusion
Tex syntax error
is a microbundle isomorphism - If
Tex syntax error
is any other such neighbourhood ofTex syntax error
then there is aTex syntax error
-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