Microbundle

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1 Definition

The concept of a microbundle of dimension $n$${{Stub}}{{Authors|Matthias Kreck}} == Definition == <wikitex>; The concept of a '''microbundle''' of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[Wikipedia:Topological manifold|topological manifold]]. Later \cite{Kister1964} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle; i.e. a fibre bundle with structure group the homeomorphisms of $\Rr^n$ fixing <script type="math/tex">n$ was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later [Kister1964] showed that every microbundle uniquely determines a topological $\Rr^n$$\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$$B$ be a topological space. An $n$$n$-dimensional microbundle over $B$$B$ is a quadruple $(E,B,i,j)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vyc85M$$(E,B,i,j)$ where $E$$E$ is a space, $i$$i$ and $j$$j$ are maps fitting into the following diagram

$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_5AGNzC$ $$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$$

and the following conditions hold:

1. $j\circ i=\id_B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1RsEts$$j\circ i=\id_B$.
2. For all $x\in B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_T4mrOi$$x\in B$ there exist open neigbourhood $U\subset B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_30TSy9$$U\subset B$, an open neighbourhood $V\subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_bcMVI0$$V\subset E$ of $i(b)$$i(b)$ and a homeomorphism
   $\displaystyle h \colon V \to U\times \mathbb{R}^n/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_dnkPjS$ $$\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$$E$ is called the total space of the bundle and $B$$B$ the base space.

Two microbundles $(E_n,B,i_n,j_n)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_HT0ylK$$(E_n,B,i_n,j_n)$, $n=1,2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_LENHMC$$n=1,2$ over the same space $B$$B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_jhJXEv$$V_1\subset E_1$ of $i_1(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_TZcUWo$$i_1(B)$ and $V_2\subset E_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_B9j3Ei$$V_2\subset E_2$ of $i_2(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_n7sz8c$$i_2(B)$ and a homeomorphism $H\colon V_1\to V_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_tQFc27$$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$$PL$) manifold $M$$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$$M$ be topological (or PL) $n$$n$-manifold, and let $p_1 \colon M \times M \to M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1qeDq3$$p_1 \colon M \times M \to M$ be the projection onto the first factor. Then

$\displaystyle (M \times M, M, \Delta_M, p_1)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_puYEfZ$ $$\displaystyle (M \times M, M, \Delta_M, p_1)$$

is an $n$$n$-dimensional microbundle, the tangent microbundle $\tau_M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_t8PwvV$$\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 $\Rr^n$$\Rr^n$-bundle.

Example 2.3. Let $\pi \colon E \to B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_53o2bS$$\pi \colon E \to B$ be a topological $\Rr^n$$\Rr^n$-bundle with zero section $s \colon B \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_jFQpiP$$s \colon B \to E$. Then the quadruple

$\displaystyle (E, B, s, \pi)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_JxnXQM$ $$\displaystyle (E, B, s, \pi)$$

is an $n$$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 $(E, B, i, j)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_tHNySK$$(E, B, i, j)$ be an $n$$n$-dimensional microbundle over a locally finite, finite dimensional simplicial complex $B$$B$. Then there is a neighbourhood of $i(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vZkfkJ$$i(B)$, $E_1 \subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_zFI3bI$$E_1 \subset E$ such that:

1. $E_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_RyrmBH$$E_1$ is the total space of a topological $\Rr^n$$\Rr^n$-bundle over $B$$B$.
2. The inclusion $E_1 \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_NUj9qH$$E_1 \to E$ is a microbundle isomorphism.
3. If $E_2 \subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_naMOJH$$E_2 \subset E$ is any other such neighbourhood of $i(B)$$i(B)$ then there is a $\Rr^n$$\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_H4rVsI$$(E_1 \to B) \cong (E_2 \to B)$.

Remark 2.5. Microbundle theory is an important part of the Kirby and Siebenmann [Kirby&Siebenmann1977] work on smooth structures and $PL$$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:}} [[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$$B$ be a topological space. An $n$$n$-dimensional microbundle over $B$$B$ is a quadruple $(E,B,i,j)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vyc85M$$(E,B,i,j)$ where $E$$E$ is a space, $i$$i$ and $j$$j$ are maps fitting into the following diagram

$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_5AGNzC$ $$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$$

and the following conditions hold:

1. $j\circ i=\id_B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1RsEts$$j\circ i=\id_B$.
2. For all $x\in B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_T4mrOi$$x\in B$ there exist open neigbourhood $U\subset B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_30TSy9$$U\subset B$, an open neighbourhood $V\subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_bcMVI0$$V\subset E$ of $i(b)$$i(b)$ and a homeomorphism
   $\displaystyle h \colon V \to U\times \mathbb{R}^n/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_dnkPjS$ $$\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$$E$ is called the total space of the bundle and $B$$B$ the base space.

Two microbundles $(E_n,B,i_n,j_n)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_HT0ylK$$(E_n,B,i_n,j_n)$, $n=1,2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_LENHMC$$n=1,2$ over the same space $B$$B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_jhJXEv$$V_1\subset E_1$ of $i_1(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_TZcUWo$$i_1(B)$ and $V_2\subset E_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_B9j3Ei$$V_2\subset E_2$ of $i_2(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_n7sz8c$$i_2(B)$ and a homeomorphism $H\colon V_1\to V_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_tQFc27$$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$$PL$) manifold $M$$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$$M$ be topological (or PL) $n$$n$-manifold, and let $p_1 \colon M \times M \to M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1qeDq3$$p_1 \colon M \times M \to M$ be the projection onto the first factor. Then

$\displaystyle (M \times M, M, \Delta_M, p_1)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_puYEfZ$ $$\displaystyle (M \times M, M, \Delta_M, p_1)$$

is an $n$$n$-dimensional microbundle, the tangent microbundle $\tau_M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_t8PwvV$$\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 $\Rr^n$$\Rr^n$-bundle.

Example 2.3. Let $\pi \colon E \to B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_53o2bS$$\pi \colon E \to B$ be a topological $\Rr^n$$\Rr^n$-bundle with zero section $s \colon B \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_jFQpiP$$s \colon B \to E$. Then the quadruple

$\displaystyle (E, B, s, \pi)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_JxnXQM$ $$\displaystyle (E, B, s, \pi)$$

is an $n$$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 $(E, B, i, j)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_tHNySK$$(E, B, i, j)$ be an $n$$n$-dimensional microbundle over a locally finite, finite dimensional simplicial complex $B$$B$. Then there is a neighbourhood of $i(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vZkfkJ$$i(B)$, $E_1 \subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_zFI3bI$$E_1 \subset E$ such that:

1. $E_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_RyrmBH$$E_1$ is the total space of a topological $\Rr^n$$\Rr^n$-bundle over $B$$B$.
2. The inclusion $E_1 \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_NUj9qH$$E_1 \to E$ is a microbundle isomorphism.
3. If $E_2 \subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_naMOJH$$E_2 \subset E$ is any other such neighbourhood of $i(B)$$i(B)$ then there is a $\Rr^n$$\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_H4rVsI$$(E_1 \to B) \cong (E_2 \to B)$.

Remark 2.5. Microbundle theory is an important part of the Kirby and Siebenmann [Kirby&Siebenmann1977] work on smooth structures and $PL$$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.