Microbundle
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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: | ||
$$ | $$ | ||
− | \xymatrix{ | + | \xymatrix{ |
+ | & V_1 \ar[dd]^H\ar[rd]^{j_1|_{V_1}} \\ | ||
+ | B\ar[ru]^{i_1}\ar[rd]_{i_2} & & B \\ | ||
+ | & V_2 \ar[ru]_{j_2|_{V_2}} | ||
+ | } | ||
$$ | $$ | ||
{{endthm|Definition}} | {{endthm|Definition}} | ||
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# If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)$. | # If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)$. | ||
{{endthm}} | {{endthm}} | ||
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</wikitex> | </wikitex> | ||
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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.
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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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.
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