Microbundle
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{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
− | == | + | == The tangent microbundle == |
<wikitex>; | <wikitex>; | ||
− | + | The most important example of a microbundle is the '''micro tangent bundle''' of a topological (or similarly $PL$) manifold $M$. | |
− | $$\ | + | Let |
− | + | $$\Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~$$ | |
− | + | be the diagonal map for $M$. | |
{{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}} | {{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}} | ||
− | Let $M$ be topological $n$-manifold, and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then | + | Let $M$ be topological (or PL) $n$-manifold, and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then |
$$ (M \times M, M, \Delta_M, p_1) $$ | $$ (M \times M, M, \Delta_M, p_1) $$ | ||
is an $n$-dimensional microbundle, the '''tangent microbundle''' $\tau_M$ of $M$. | is an $n$-dimensional microbundle, the '''tangent microbundle''' $\tau_M$ of $M$. | ||
{{endrem}} | {{endrem}} | ||
− | {{beginrem|Example}} Let $\pi \colon E \to B$ be a topological $\Rr^n$-bundle with zero section $s \colon B \to E$. Then $$(E, B, s, \pi)$$ | + | |
+ | {{beginrem|Remark}} | ||
+ | An atlas of $M$ gives a product atlas of $M \times M$ which shows that the second condition of a microbundle is fulfilled. Actually the definition of the micro tangent bundle looks a bit more like a normal bundle to the diagonal, a view which fits to the fact that the normal bundle of the diagonal of a smooth manifold $M$ in $M \times M$ is isomorphic to its tangent bundle. | ||
+ | {{endrem}} | ||
+ | |||
+ | Another important example of a microbundle is the micro-bundle defined by a topological topological $\Rr^n$-bundle. | ||
+ | |||
+ | {{beginrem|Example}} Let $\pi \colon E \to B$ be a topological $\Rr^n$-bundle with zero section $s \colon B \to E$. | ||
+ | Then | ||
+ | $$(E, B, s, \pi)$$ | ||
is an $n$-dimensional microbundle. | is an $n$-dimensional microbundle. | ||
{{endrem}} | {{endrem}} | ||
+ | A fundamental theorem about microbundles is the following theorem, often called the Kister-Mazur theorem. | ||
{{beginthm|Theorem|\cite{Kister1964|Theorem 2}}} | {{beginthm|Theorem|\cite{Kister1964|Theorem 2}}} | ||
− | Let $(E, B, i, j)$ be an $n$-dimensional microbundle. | + | Let $(E, B, i, j)$ be an $n$-dimensional microbundle over a locally finite, finite dimensional simplicial complex $B$. |
+ | 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}} | ||
+ | {{beginrem|Remark}} | ||
+ | Microbundle theory is an important part of the Kirby and Siebenmann {{cite|Kirby&Siebenmann1977}} work on smooth structures and $PL$-structures on higher dimensional manifolds. | ||
+ | {{endrem}} | ||
</wikitex> | </wikitex> | ||
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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
The most important example of a microbundle is the micro tangent bundle 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
is an -dimensional microbundle.
A fundamental theorem 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
The most important example of a microbundle is the micro tangent bundle 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
is an -dimensional microbundle.
A fundamental theorem 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