Microbundle

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.

This page is being independently reviewed under the supervision of the Editorial Board.

1 Definition

The concept of a microbundle of dimension

Tex syntax error

${{Authors|Diarmuid Crowley|Matthias Kreck}}{{Definition reviewed}} ==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 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 <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 Kister [Kister1964], and independently Mazur, showed that every microbundle uniquely determines a topological

Tex syntax error

$\Rr^n$-bundle; i.e. a fibre bundle with structure group the homeomorphisms of

Tex syntax error

$\Rr^n$ fixing $0$$0$.

Definition 1.1 [Milnor1964] .

Let

Tex syntax error

$B$ be a topological space. An

Tex syntax error

$n$-dimensional microbundle over

Tex syntax error

$B$ is a quadruple

Tex syntax error

$(E,B,i,j)$

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

Tex syntax error

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

and the following conditions hold:

1. Tex syntax error

   $j\circ i=\id_B$.
2. For all

   Tex syntax error

   $x\in B$ there exist open neigbourhood

   Tex syntax error

   $U\subset B$, an open neighbourhood

   Tex syntax error

   $V\subset E$ of

   Tex syntax error

   $i(b)$ and a homeomorphism

   Tex syntax error

   $$\displaystyle h \colon V \to U\times \mathbb{R}^n$$

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$$E$ is called the total space of the bundle and

Tex syntax error

$B$ 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

Tex syntax error

$B$ 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 Examples

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

Example 2.1 [Milnor1964, Lemma 2.1].

Let $M$$M$ be topological (or PL)

Tex syntax error

$n$-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

Tex syntax error

$n$-dimensional microbundle, the tangent microbundle

Tex syntax error

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

Remark 2.2. An atlas of $M$$M$ gives a product atlas of $M \times M$$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$$M$ in $M \times M$$M \times M$ is isomorphic to its tangent bundle.

Another important example of a microbundle is the micro-bundle defined by a topological topological

Tex syntax error

$\Rr^n$-bundle.

Example 2.3. Let

Tex syntax error

$\pi \colon E \to B$ be a topological

Tex syntax error

$\Rr^n$-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

Tex syntax error

$n$-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

Tex syntax error

$(E, B, i, j)$ be an

Tex syntax error

$n$-dimensional microbundle over a locally finite, finite dimensional simplicial complex

Tex syntax error

$B$. Then there is a neighbourhood of

Tex syntax error

$i(B)$,

Tex syntax error

$E_1 \subset E$ such that the following hold.

1. Tex syntax error

   $E_1$ is the total space of a topological

   Tex syntax error

   $\Rr^n$-bundle over

   Tex syntax error

   $B$.
2. $(E_1, B, i, j|_{E_1})$$(E_1, B, i, j|_{E_1})$ is a microbundle and the the inclusion

   Tex syntax error

   $E_1 \to E$ is a microbundle isomorphism.
3. If

   Tex syntax error

   $E_2 \subset E$ is any other such neighbourhood of

   Tex syntax error

   $i(B)$ then there is a

   Tex syntax error

   $\Rr^n$-bundle isomorphism $(E_1, B, i, j|_{E_1}) \cong (E_2, B, i, j|_{E_2})$$(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$$PL$-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.

$. {{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}} == Examples == ; 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}} == 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. {{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 the following hold. # $E_1$ 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 $(E_1, B, i, j|_{E_1}) \cong (E_2, B, i, j|_{E_2})$. {{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}} == References == {{#RefList:}} == External links == * The Wikipedia page about [[Wikipedia:Microbundle|microbundles]]. [[Category:Definitions]]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

Tex syntax error

$\Rr^n$-bundle; i.e. a fibre bundle with structure group the homeomorphisms of

Tex syntax error

$\Rr^n$ fixing $0$$0$.

Definition 1.1 [Milnor1964] .

Let

Tex syntax error

$B$ be a topological space. An

Tex syntax error

$n$-dimensional microbundle over

Tex syntax error

$B$ is a quadruple

Tex syntax error

$(E,B,i,j)$

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

Tex syntax error

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

and the following conditions hold:

1. Tex syntax error

   $j\circ i=\id_B$.
2. For all

   Tex syntax error

   $x\in B$ there exist open neigbourhood

   Tex syntax error

   $U\subset B$, an open neighbourhood

   Tex syntax error

   $V\subset E$ of

   Tex syntax error

   $i(b)$ and a homeomorphism

   Tex syntax error

   $$\displaystyle h \colon V \to U\times \mathbb{R}^n$$

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$$E$ is called the total space of the bundle and

Tex syntax error

$B$ 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

Tex syntax error

$B$ 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 Examples

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

Example 2.1 [Milnor1964, Lemma 2.1].

Let $M$$M$ be topological (or PL)

Tex syntax error

$n$-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

Tex syntax error

$n$-dimensional microbundle, the tangent microbundle

Tex syntax error

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

Remark 2.2. An atlas of $M$$M$ gives a product atlas of $M \times M$$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$$M$ in $M \times M$$M \times M$ is isomorphic to its tangent bundle.

Another important example of a microbundle is the micro-bundle defined by a topological topological

Tex syntax error

$\Rr^n$-bundle.

Example 2.3. Let

Tex syntax error

$\pi \colon E \to B$ be a topological

Tex syntax error

$\Rr^n$-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

Tex syntax error

$n$-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

Tex syntax error

$(E, B, i, j)$ be an

Tex syntax error

$n$-dimensional microbundle over a locally finite, finite dimensional simplicial complex

Tex syntax error

$B$. Then there is a neighbourhood of

Tex syntax error

$i(B)$,

Tex syntax error

$E_1 \subset E$ such that the following hold.

1. Tex syntax error

   $E_1$ is the total space of a topological

   Tex syntax error

   $\Rr^n$-bundle over

   Tex syntax error

   $B$.
2. $(E_1, B, i, j|_{E_1})$$(E_1, B, i, j|_{E_1})$ is a microbundle and the the inclusion

   Tex syntax error

   $E_1 \to E$ is a microbundle isomorphism.
3. If

   Tex syntax error

   $E_2 \subset E$ is any other such neighbourhood of

   Tex syntax error

   $i(B)$ then there is a

   Tex syntax error

   $\Rr^n$-bundle isomorphism $(E_1, B, i, j|_{E_1}) \cong (E_2, B, i, j|_{E_2})$$(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$$PL$-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.