Microbundle

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An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print.

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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 [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 be a topological space. An n-dimensional microbundle over B is a quadruple (E,B,i,j)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vyc85M where E is a space, i and 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

and the following conditions hold:

  1. j\circ i=\id_B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1RsEts.
  2. For all x\in B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_T4mrOi there exist open neigbourhood U\subset B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_30TSy9, an open neighbourhood V\subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_bcMVI0 of 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

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)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_HT0ylK, n=1,2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_LENHMC over the same space B are isomorphic if there exist neighbourhoods V_1\subset E_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_jhJXEv of i_1(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_TZcUWo and V_2\subset E_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_B9j3Ei of i_2(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_n7sz8c and a homeomorphism H\colon V_1\to V_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_tQFc27 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)~

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/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1qeDq3 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

is an n-dimensional microbundle, the tangent microbundle \tau_M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_t8PwvV 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/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_53o2bS be a topological \Rr^n-bundle with zero section s \colon B \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_jFQpiP. Then the quadruple

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

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

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

Remark 2.5. 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

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-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)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vyc85M where E is a space, i and 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

and the following conditions hold:

  1. j\circ i=\id_B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1RsEts.
  2. For all x\in B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_T4mrOi there exist open neigbourhood U\subset B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_30TSy9, an open neighbourhood V\subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_bcMVI0 of 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

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)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_HT0ylK, n=1,2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_LENHMC over the same space B are isomorphic if there exist neighbourhoods V_1\subset E_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_jhJXEv of i_1(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_TZcUWo and V_2\subset E_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_B9j3Ei of i_2(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_n7sz8c and a homeomorphism H\colon V_1\to V_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_tQFc27 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)~

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/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1qeDq3 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

is an n-dimensional microbundle, the tangent microbundle \tau_M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_t8PwvV 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/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_53o2bS be a topological \Rr^n-bundle with zero section s \colon B \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_jFQpiP. Then the quadruple

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

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

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

Remark 2.5. 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

4 External links

The Wikipedia page about microbundles.
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