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The concept of 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. | The concept of 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. | ||
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{{beginthm|Definition|{{cite|Milnor1964}} }} | {{beginthm|Definition|{{cite|Milnor1964}} }} | ||
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$$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$ | $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$ | ||
{{endthm}} | {{endthm}} | ||
+ | |||
+ | For any space $M$ define the diagonal embedding | ||
+ | $$\Delta_M \colon M \to M \times M;x \mapsto (x,x)~.$$ | ||
+ | If $M$ is a differentiable $n$-manifold the normal bundle of $\nu_M$ is the tangent bundle $\tau_M$. | ||
+ | In the topological category we have: | ||
{{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}} | {{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}} | ||
− | Let $M$ be topological $n$-manifold, | + | Let $M$ be topological $n$-manifold, and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then |
$$ (M \times M, M, \Delta_M, p_1) $$ | $$ (M \times M, M, \Delta_M, p_1) $$ | ||
is an $n$-dimensional microbundle. | is an $n$-dimensional microbundle. |
Revision as of 07:04, 9 June 2012
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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 quadruple such that there is a sequence- for all there exist open neigbourhood , an open neighbourhood of and a homeomorphism
which makes the following diagram commute:
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For any space define the diagonal embedding
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If is a differentiable -manifold the normal bundle of is the tangent bundle . 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.
is an -dimensional microbundle.
Definition 1.4. Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute.
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Theorem 1.5 [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