Microbundle
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− | {{Stub}} | + | {{Stub}}{{Authors|Matthias Krec}} |
== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | The concept of a | + | The concept of a '''microbundle''' 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; i.e. a fibre bundle with structure group the homeomorphisms of $\Rr^n$ fixing $0$. |
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{{beginthm|Definition|{{cite|Milnor1964}} }} | {{beginthm|Definition|{{cite|Milnor1964}} }} | ||
− | An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ | + | Let $B$ be a topological space. An '''$n$-dimensional microbundle''' over $B$ is a quadruple $(E,B,i,j)$ |
+ | where $E$ is a space, $i$ and $j$ are maps fitting into the following diagram | ||
+ | $$B\xrightarrow{i} E\xrightarrow{j} B$$ | ||
+ | and the following conditions hold: | ||
#$j\circ i=\id_B$ | #$j\circ i=\id_B$ | ||
#for all $x\in B$ there exist open neigbourhood $U\subset B$, an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n$$ | #for all $x\in B$ there exist open neigbourhood $U\subset B$, an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n$$ | ||
which makes the following diagram commute: | which makes the following diagram commute: | ||
− | $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$ | + | $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}. $$ |
+ | The space $E$ is called the '''total space''' of the bundle and $B$ the '''base space'''. | ||
+ | |||
+ | 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: | ||
+ | $$ | ||
+ | \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \\ | ||
+ | B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \\ | ||
+ | & V_2 \ar[ur]_{j_2|_{V_2}}. } | ||
+ | $$ | ||
{{endthm}} | {{endthm}} | ||
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{{endrem}} | {{endrem}} | ||
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{{beginthm|Theorem|\cite{Kister1964|Theorem 2}}} | {{beginthm|Theorem|\cite{Kister1964|Theorem 2}}} |
Revision as of 19:21, 22 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. |
This page has not been refereed. The information given here might be incomplete or provisional. |
The user responsible for this page is Matthias Krec. No other user may edit this page at present. |
1 Introduction
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:
For any space define the diagonal embedding
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Tex syntax erroris the tangent bundle of .
In the topological category we have:
Example 1.2 [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 1.4 [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
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.
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