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} | + | \xymatrix{ & V_1 \ar[dd]^H \arrd]^{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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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 quadrupleTex syntax errorsuch that there is a sequence
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Tex syntax error
- for all
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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:
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For any space define the diagonal embedding
Tex syntax error
Tex syntax erroris the tangent bundle
Tex syntax errorof .
In the topological category we have:
Example 1.2 [Milnor1964, Lemma 2.1].
Let be topological -manifold, and letTex syntax errorbe the projection onto the first factor. Then
Tex syntax error
Tex syntax errorof .
Tex syntax errorbe a topological -bundle with zero section
Tex syntax error. Then
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is an -dimensional microbundle.
Definition 1.4.
Two microbundlesTex syntax error,
Tex syntax errorover the same space are 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:
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Theorem 1.5 [Kister1964, Theorem 2] .
LetTex syntax errorbe an -dimensional microbundle. Then there is a neighbourhood of
Tex syntax error,
Tex syntax errorsuch that:
-
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is the total space of a topological -bundle over . - The inclusion
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is a microbundle isomorphism - If
Tex syntax error
is any other such neighbourhood ofTex syntax error
then there is a -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