Microbundle
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− | {{Stub}}{{Authors|Matthias | + | {{Stub}}{{Authors|Matthias Kreck}} |
== Definition == | == Definition == | ||
<wikitex>; | <wikitex>; | ||
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$$B\xrightarrow{i} E\xrightarrow{j} B$$ | $$B\xrightarrow{i} E\xrightarrow{j} B$$ | ||
and the following conditions hold: | 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 $$h \colon V \to U\times \mathbb{R}^n$$ |
which makes the following diagram commute: | which makes the following diagram commute: | ||
$$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}. $$ | $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}. $$ | ||
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\xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \\ | \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}} } |
$$ | $$ | ||
{{endthm}} | {{endthm}} | ||
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== The tangent microbundle == | == The tangent microbundle == | ||
<wikitex>; | <wikitex>; | ||
− | + | An important example of a microbundle is the '''tangent microbundle''' of a topological (or similarly $PL$) manifold $M$. | |
Let | Let | ||
$$\Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~$$ | $$\Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~$$ | ||
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{{beginrem|Example}} Let $\pi \colon E \to B$ be a topological $\Rr^n$-bundle with zero section $s \colon B \to E$. | {{beginrem|Example}} Let $\pi \colon E \to B$ be a topological $\Rr^n$-bundle with zero section $s \colon B \to E$. | ||
− | Then | + | Then the quadruple |
$$(E, B, s, \pi)$$ | $$(E, B, s, \pi)$$ | ||
is an $n$-dimensional microbundle. | is an $n$-dimensional microbundle. | ||
{{endrem}} | {{endrem}} | ||
− | A fundamental | + | A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem. |
{{beginthm|Theorem|\cite{Kister1964|Theorem 2}}} | {{beginthm|Theorem|\cite{Kister1964|Theorem 2}}} | ||
Line 53: | Line 53: | ||
Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that: | 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 $B$. | ||
− | # The inclusion $E_1 \to E$ is a microbundle isomorphism | + | # The inclusion $E_1 \to E$ is a microbundle isomorphism. |
# 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}} |
Revision as of 19:44, 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 Kreck. No other user may edit this page at present. |
1 Definition
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:
2 The tangent microbundle
An important example of a microbundle is the tangent microbundle of a topological (or similarly ) manifold . Let
be the diagonal map for .
Example 2.1 [Milnor1964, Lemma 2.1]. Let be topological (or PL) -manifold, and let be the projection onto the first factor. Then
is an -dimensional microbundle, the tangent microbundle of .
Remark 2.2. An atlas of gives a product atlas of which shows that the second condition of a microbundle is fulfilled. Actually the definition of the micro tangent bundle looks a bit more like a normal bundle to the diagonal, a view which fits to the fact that the normal bundle of the diagonal of a smooth manifold in is isomorphic to its tangent bundle.
Another important example of a microbundle is the micro-bundle defined by a topological topological -bundle.
Example 2.3. Let be a topological -bundle with zero section . Then the quadruple
is an -dimensional microbundle.
A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem.
Theorem 2.4 [Kister1964, Theorem 2]. Let be an -dimensional microbundle over a locally finite, finite dimensional simplicial complex . 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 .
Remark 2.5. Microbundle theory is an important part of the Kirby and Siebenmann [Kirby&Siebenmann1977] work on smooth structures and -structures on higher dimensional manifolds.
3 References
- [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004
- [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
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:
2 The tangent microbundle
An important example of a microbundle is the tangent microbundle of a topological (or similarly ) manifold . Let
be the diagonal map for .
Example 2.1 [Milnor1964, Lemma 2.1]. Let be topological (or PL) -manifold, and let be the projection onto the first factor. Then
is an -dimensional microbundle, the tangent microbundle of .
Remark 2.2. An atlas of gives a product atlas of which shows that the second condition of a microbundle is fulfilled. Actually the definition of the micro tangent bundle looks a bit more like a normal bundle to the diagonal, a view which fits to the fact that the normal bundle of the diagonal of a smooth manifold in is isomorphic to its tangent bundle.
Another important example of a microbundle is the micro-bundle defined by a topological topological -bundle.
Example 2.3. Let be a topological -bundle with zero section . Then the quadruple
is an -dimensional microbundle.
A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem.
Theorem 2.4 [Kister1964, Theorem 2]. Let be an -dimensional microbundle over a locally finite, finite dimensional simplicial complex . 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 .
Remark 2.5. Microbundle theory is an important part of the Kirby and Siebenmann [Kirby&Siebenmann1977] work on smooth structures and -structures on higher dimensional manifolds.
3 References
- [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004
- [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