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{ & V_1 \ar[dd]^H \ | + | \xymatrix{ & V_1 \ar[dd]^H \ar[dr]^{j_1|_{V_1} \\ |
B \ar[ur]^{i_1} ar[dr]_{i_2} && B \\ | B \ar[ur]^{i_1} ar[dr]_{i_2} && B \\ | ||
& V_2 \ar[ur]_{j_2|_{V_2} } | & V_2 \ar[ur]_{j_2|_{V_2} } | ||
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{{endthm|Definition}} | {{endthm|Definition}} | ||
− | {{beginthm|Theorem|\cite{Kister1964|Theorem 2} }} | + | {{beginthm|Theorem|\cite{Kister1964|Theorem 2}}} |
Let $(E, B, i, j)$ be an $n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that: | Let $(E, B, i, j)$ be an $n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that: | ||
# $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$. | # $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$. |
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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:
For any space define the diagonal embedding
If is a differentiable -manifold the normal bundle of is 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.
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 .
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