Microbundle
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− | {{ | + | {{Authors|Diarmuid Crowley|Matthias Kreck}} |
− | == | + | ==Definition== |
<wikitex>; | <wikitex>; | ||
− | The concept of a | + | 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 $0$. |
− | + | ||
− | + | ||
{{beginthm|Definition|{{cite|Milnor1964}} }} | {{beginthm|Definition|{{cite|Milnor1964}} }} | ||
− | An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ | + | Let $B$ be a topological space. An '''$n$-dimensional microbundle''' over $B$ is a quadruple $(E,B,i,j)$ |
− | #$j\circ i=\id_B$ | + | 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: | which makes the following diagram commute: | ||
− | $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$ | + | $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}. $$ |
+ | 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}} | {{endthm}} | ||
+ | </wikitex> | ||
− | + | == Examples == | |
− | $$\Delta_M \colon M \to M \times M | + | <wikitex>; |
− | + | An important example of a microbundle is the '''tangent microbundle''' of a topological (or similarly $PL$) manifold $M$. | |
− | + | Let | |
+ | $$\Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~$$ | ||
+ | be the diagonal map for $M$. | ||
{{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}} | {{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}} | ||
− | Let $M$ be topological $n$-manifold, and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then | + | Let $M$ be topological (or PL) $n$-manifold, and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then |
$$ (M \times M, M, \Delta_M, p_1) $$ | $$ (M \times M, M, \Delta_M, p_1) $$ | ||
− | is an $n$-dimensional microbundle, the | + | is an $n$-dimensional microbundle, the '''tangent microbundle''' $\tau_M$ of $M$. |
{{endrem}} | {{endrem}} | ||
− | {{beginrem| | + | |
− | is | + | {{beginrem|Remark}} |
+ | An atlas of $M$ gives a product atlas of $M \times M$ which shows that the second condition of a microbundle is fulfilled. Actually the definition of the micro tangent bundle looks a bit more like a normal bundle to the diagonal, a view which fits to the fact that the normal bundle of the diagonal of a smooth manifold $M$ in $M \times M$ is isomorphic to its tangent bundle. | ||
{{endrem}} | {{endrem}} | ||
− | {{ | + | Another important example of a microbundle is the micro-bundle defined by a topological topological $\Rr^n$-bundle. |
− | + | ||
− | $$ | + | {{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}} | |
− | + | </wikitex> | |
− | + | == The Kister-Mazur Theorem == | |
+ | <wikitex>; | ||
+ | A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem, proven independently by Kister and Mazur. | ||
{{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 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 $B$. | ||
− | # | + | # $(E_1, B, i, j|_{E_1})$ 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 | + | # 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}} | ||
+ | {{beginrem|Remark}} | ||
+ | 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. | ||
+ | {{endrem}} | ||
</wikitex> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
− | [[Category: | + | == External links == |
+ | * The Wikipedia page about [[Wikipedia:Microbundle|microbundles]]. | ||
+ | |||
+ | [[Category:Definitions]] |
Latest revision as of 14:20, 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 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 -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 Examples
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.
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 be an -dimensional microbundle over a locally finite, finite dimensional simplicial complex . Then there is a neighbourhood of , such that the following hold.
- is the total space of a topological -bundle over .
- is a microbundle and the the inclusion is a microbundle isomorphism.
- If is any other such neighbourhood of then there is a -bundle isomorphism .
Remark 3.2. Microbundle theory is an important part of the work by Kirby and Siebenmann [Kirby&Siebenmann1977] on smooth structures and -structures on higher dimensional topological manifolds.
4 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
5 External links
- The Wikipedia page about microbundles.