Microbundle

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1 Definition

The concept of a microbundle of dimension ${{Stub}}{{Authors|Matthias Kreck}} == Definition == <wikitex>; The concept of a '''microbundle''' of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[Wikipedia:Topological manifold|topological manifold]]. Later \cite{Kister1964} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle; i.e. a fibre bundle with structure group the homeomorphisms of $\Rr^n$ fixing <script type="math/tex">n$ 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 $\Rr^n$ -bundle; i.e. a fibre bundle with structure group the homeomorphisms of $\Rr^n$ fixing $0$ .

Definition 1.1 [Milnor1964] .

Let $B$ $B$ be a topological space. An $n$ -dimensional microbundle over $B$ $B$ is a quadruple

Tex syntax error

$(E,B,i,j)$

where $E$ is a space, $i$ and $j$ are maps fitting into the following diagram

Tex syntax error

$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$

and the following conditions hold:

```
Tex syntax error
```
.
For all
```
Tex syntax error
```
there exist open neigbourhood
```
Tex syntax error
```
, an open neighbourhood
```
Tex syntax error
```
of
```
Tex syntax error
```
and a homeomorphism
```
Tex syntax error
```

which makes the following diagram commute:

$\displaystyle \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}.$

The space $E$ is called the total space of the bundle and $B$ the base space.

Two microbundles

Tex syntax error

$(E_n,B,i_n,j_n)$ ,

Tex syntax error

$n=1,2$ over the same space $B$ $B$ are isomorphic if there exist neighbourhoods

Tex syntax error

$V_1\subset E_1$ of

Tex syntax error

$i_1(B)$ and

Tex syntax error

$V_2\subset E_2$ of

Tex syntax error

$i_2(B)$ and a homeomorphism

Tex syntax error

$H\colon V_1\to V_2$ making the following diagram commute:

$\displaystyle \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \\ B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \\ & V_2 \ar[ur]_{j_2|_{V_2}} }$

2 The tangent microbundle

An important example of a microbundle is the tangent microbundle of a topological (or similarly $PL$ ) manifold $M$ . Let

$\displaystyle \Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~$ $\displaystyle \Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~$

be the diagonal map for $M$ .

Example 2.1 [Milnor1964, Lemma 2.1].

Let $M$ $M$ be topological (or PL) $n$ $n$ -manifold, and let

Tex syntax error

$p_1 \colon M \times M \to M$ be the projection onto the first factor. Then

Tex syntax error

$\displaystyle (M \times M, M, \Delta_M, p_1)$

is an $n$ $n$ -dimensional microbundle, the tangent microbundle

Tex syntax error

$\tau_M$ of $M$ $M$ .

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.

Another important example of a microbundle is the micro-bundle defined by a topological topological $\Rr^n$ -bundle.

Example 2.3. Let

Tex syntax error

$\pi \colon E \to B$ be a topological $\Rr^n$ $\Rr^n$ -bundle with zero section

Tex syntax error

$s \colon B \to E$ .

Then the quadruple

Tex syntax error

$\displaystyle (E, B, s, \pi)$

is an $n$ -dimensional microbundle.

A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem.

Theorem 2.4 [Kister1964, Theorem 2].

Let

Tex syntax error

$(E, B, i, j)$ be an $n$ $n$ -dimensional microbundle over a locally finite, finite dimensional simplicial complex $B$ $B$ . Then there is a neighbourhood of

Tex syntax error

$i(B)$ ,

Tex syntax error

$E_1 \subset E$ such that:

```
Tex syntax error
```
is the total space of a topological $\Rr^n$ -bundle over $B$ .
The inclusion
```
Tex syntax error
```
is a microbundle isomorphism.
If
```
Tex syntax error
```
is any other such neighbourhood of
```
Tex syntax error
```
then there is a $\Rr^n$ -bundle isomorphism
```
Tex syntax error
```
.

Microbundle theory is an important part of the Kirby and Siebenmann [Kirby&Siebenmann1977] work on smooth structures and $PL$ -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

4 External links

The Wikipedia page about microbundles.

$. {{beginthm|Definition|{{cite|Milnor1964}} }} Let $B$ be a topological space. An '''$n$-dimensional microbundle''' over $B$ is a quadruple $(E,B,i,j)$ where $E$ is a space, $i$ and $j$ are maps fitting into the following diagram $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold: #$j\circ i=\id_B$. #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: $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \ & U \times \Rr^n \ar[ur]_{p_1}}. $$ The space $E$ is called the '''total space''' of the bundle and $B$ the '''base space'''. 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: $$ \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \ B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \ & V_2 \ar[ur]_{j_2|_{V_2}} } $$ {{endthm}} == The tangent microbundle == ; An important example of a microbundle is the '''tangent microbundle''' 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}}}} 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 $$ (M \times M, M, \Delta_M, p_1) $$ is an $n$-dimensional microbundle, the '''tangent microbundle''' $\tau_M$ of $M$. {{endrem}} {{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 the quadruple $$(E, B, s, \pi)$$ is an $n$-dimensional microbundle. {{endrem}} A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem. {{beginthm|Theorem|\cite{Kister1964|Theorem 2}}} 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 $B$. # 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 $(E_1 \to B) \cong (E_2 \to B)$. {{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}} == References == {{#RefList:}} == External links == The Wikipedia page about [[Wikipedia:Microbundle|microbundles]]. [[Category:Theory]]n 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 $\Rr^n$ $\Rr^n$ -bundle; i.e. a fibre bundle with structure group the homeomorphisms of $\Rr^n$ $\Rr^n$ fixing $0$ $0$ .

Definition 1.1 [Milnor1964] .

Let $B$ $B$ be a topological space. An $n$ -dimensional microbundle over $B$ $B$ is a quadruple

Tex syntax error

$(E,B,i,j)$

where $E$ is a space, $i$ and $j$ are maps fitting into the following diagram

Tex syntax error

$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$

and the following conditions hold:

```
Tex syntax error
```
.
For all
```
Tex syntax error
```
there exist open neigbourhood
```
Tex syntax error
```
, an open neighbourhood
```
Tex syntax error
```
of
```
Tex syntax error
```
and a homeomorphism
```
Tex syntax error
```

which makes the following diagram commute:

$\displaystyle \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}.$

The space $E$ is called the total space of the bundle and $B$ the base space.

Two microbundles

Tex syntax error

$(E_n,B,i_n,j_n)$ ,

Tex syntax error

$n=1,2$ over the same space $B$ $B$ are isomorphic if there exist neighbourhoods

Tex syntax error

$V_1\subset E_1$ of

Tex syntax error

$i_1(B)$ and

Tex syntax error

$V_2\subset E_2$ of

Tex syntax error

$i_2(B)$ and a homeomorphism

Tex syntax error

$H\colon V_1\to V_2$ making the following diagram commute:

$\displaystyle \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \\ B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \\ & V_2 \ar[ur]_{j_2|_{V_2}} }$

2 The tangent microbundle

An important example of a microbundle is the tangent microbundle of a topological (or similarly $PL$ ) manifold $M$ . Let

$\displaystyle \Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~$ $\displaystyle \Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~$

be the diagonal map for $M$ .

Example 2.1 [Milnor1964, Lemma 2.1].

Let $M$ $M$ be topological (or PL) $n$ $n$ -manifold, and let

Tex syntax error

$p_1 \colon M \times M \to M$ be the projection onto the first factor. Then

Tex syntax error

$\displaystyle (M \times M, M, \Delta_M, p_1)$

is an $n$ $n$ -dimensional microbundle, the tangent microbundle

Tex syntax error

$\tau_M$ of $M$ $M$ .

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.

Another important example of a microbundle is the micro-bundle defined by a topological topological $\Rr^n$ -bundle.

Example 2.3. Let

Tex syntax error

$\pi \colon E \to B$ be a topological $\Rr^n$ $\Rr^n$ -bundle with zero section

Tex syntax error

$s \colon B \to E$ .

Then the quadruple

Tex syntax error

$\displaystyle (E, B, s, \pi)$

is an $n$ -dimensional microbundle.

A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem.

Theorem 2.4 [Kister1964, Theorem 2].

Let

Tex syntax error

$(E, B, i, j)$ be an $n$ $n$ -dimensional microbundle over a locally finite, finite dimensional simplicial complex $B$ $B$ . Then there is a neighbourhood of

Tex syntax error

$i(B)$ ,

Tex syntax error

$E_1 \subset E$ such that:

```
Tex syntax error
```
is the total space of a topological $\Rr^n$ -bundle over $B$ .
The inclusion
```
Tex syntax error
```
is a microbundle isomorphism.
If
```
Tex syntax error
```
is any other such neighbourhood of
```
Tex syntax error
```
then there is a $\Rr^n$ -bundle isomorphism
```
Tex syntax error
```
.

Microbundle theory is an important part of the Kirby and Siebenmann [Kirby&Siebenmann1977] work on smooth structures and $PL$ -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

4 External links

The Wikipedia page about microbundles.

Microbundle

Revision as of 19:54, 22 December 2012

Contents

1 Definition

2 The tangent microbundle

3 References

4 External links

2 The tangent microbundle

3 References

4 External links

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@@ Line 63: / Line 63: @@
 == References ==
 {{#RefList:}}
+== External links ==
+ The Wikipedia page about [[Wikipedia:Microbundle|microbundles]].
 [[Category:Theory]]