Microbundle

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== Introduction ==
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{{Authors|Diarmuid Crowley|Matthias Kreck}}
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==Definition==
<wikitex>;
<wikitex>;
The concept of Microbundle of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[topological manifold]]. Later \cite{Kister1964} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle.
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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 Kister \cite{Kister1964}, and independently Mazur, 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 $0$.
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{{beginthm|Definition|{{cite|Milnor1964}} }}
{{beginthm|Definition|{{cite|Milnor1964}} }}
An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold.
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Let $B$ be a topological space. An '''$n$-dimensional microbundle''' over $B$ is a quadruple $(E,B,i,j)$
#$j\circ i=\id_B$
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where $E$ is a space, $i$ and $j$ are maps fitting into the following diagram
#for all $x\in B$ there exist open neigbourhood $U\subset B$ and an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n.$$
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$$B\xrightarrow{i} E\xrightarrow{j} B$$
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and the following conditions hold:
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#$j\circ i=\id_B$.
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#For all $x\in B$ there exist open neigbourhood $U\subset B$, an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n$$
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which makes the following diagram commute:
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$$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}. $$
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The space $E$ is called the '''total space''' of the bundle and $B$ the '''base space'''.
Moreover, the homeomorphism above must make the following diagram commute:
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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:
$$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$
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$$
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\xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \\
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B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \\
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& V_2 \ar[ur]_{j_2|_{V_2}} }
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$$
{{endthm}}
{{endthm}}
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</wikitex>
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== Examples ==
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<wikitex>;
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An important example of a microbundle is the '''tangent microbundle''' of a topological (or similarly $PL$) manifold $M$.
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Let
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$$\Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~$$
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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, let $\Delta_M \colon M \to M \times M$ be the diagonal map and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then
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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.
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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 $$(E, B, s, \pi)$$
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is an $n$-dimensional microbundle.
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{{beginrem|Remark}}
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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}}
{{endrem}}
To do: definition of microbundle isomorphism.
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Another important example of a microbundle is the micro-bundle defined by a topological topological $\Rr^n$-bundle.
{{beginthm|Theorem|\cite{Kister1964|Theorem 2} }}
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{{beginrem|Example}} Let $\pi \colon E \to B$ be a topological $\Rr^n$-bundle with zero section $s \colon B \to E$. Then the quadruple
Let $(E, B, i, j)$ be an $n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that:
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$$(E, B, s, \pi)$$
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is an $n$-dimensional microbundle.
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{{endrem}}
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</wikitex>
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== The Kister-Mazur Theorem ==
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<wikitex>;
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A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem, proven independently by Kister and Mazur.
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{{beginthm|Theorem|\cite{Kister1964|Theorem 2}}}
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Let $(E, B, i, j)$ be an $n$-dimensional microbundle over a locally finite, finite dimensional simplicial complex $B$.
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Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that the following hold.
# $E_1$ 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
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# $(E_1, B, i, j|_{E_1})$ is a microbundle and the 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 $(E_1 \to B) \cong (E_2 \to B)$.
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# If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1, B, i, j|_{E_1}) \cong (E_2, B, i, j|_{E_2})$.
{{endthm}}
{{endthm}}
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{{beginrem|Remark}}
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Microbundle theory is an important part of the work by Kirby and Siebenmann {{cite|Kirby&Siebenmann1977}} on smooth structures and $PL$-structures on higher dimensional topological manifolds.
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{{endrem}}
</wikitex>
</wikitex>
== References ==
== References ==
{{#RefList:}}
{{#RefList:}}
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== External links ==
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* The Wikipedia page about [[Wikipedia:Microbundle|microbundles]].
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[[Category:Definitions]]

Latest revision as of 13:20, 16 May 2013

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.

The users responsible for this page are: Diarmuid Crowley, Matthias Kreck. No other users may edit this page at present.

Contents

1 Definition

The concept of a microbundle of dimension n was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later Kister [Kister1964], and independently Mazur, 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 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

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

and the following conditions hold:

  1. j\circ i=\id_B.
  2. For all x\in B there exist open neigbourhood U\subset B, an open neighbourhood V\subset E of i(b) and a homeomorphism
    \displaystyle h \colon V \to U\times \mathbb{R}^n

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 (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:

\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 Examples

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)~

be the diagonal map for M.

Example 2.1 [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

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

is an n-dimensional microbundle, the tangent microbundle \tau_M of M.

Remark 2.2. 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 \pi \colon E \to B be a topological \Rr^n-bundle with zero section s \colon B \to E. Then the quadruple

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

is an n-dimensional microbundle.

3 The Kister-Mazur Theorem

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

Theorem 3.1 [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 the following hold.

  1. E_1 is the total space of a topological \Rr^n-bundle over B.
  2. (E_1, B, i, j|_{E_1}) is a microbundle and the the inclusion E_1 \to E is a microbundle isomorphism.
  3. If E_2 \subset E is any other such neighbourhood of i(B) then there is a \Rr^n-bundle isomorphism (E_1, B, i, j|_{E_1}) \cong (E_2, B, i, j|_{E_2}).

Remark 3.2. Microbundle theory is an important part of the work by Kirby and Siebenmann [Kirby&Siebenmann1977] on smooth structures and PL-structures on higher dimensional topological manifolds.

4 References

5 External links

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