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.

You may view the version used for publication as of 12:20, 16 May 2013 and the changes since publication.

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1 Introduction

The concept of a microbundle of dimension ${{Stub}} == Introduction == <wikitex>; The concept of a <i>microbundle</i> 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. {{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. #$j\circ i=\id_B$ #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: $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \ & U \times \Rr^n \ar[ur]_{p_1}} $$ {{endthm}} For any space $M$ define the diagonal embedding $$\Delta_M \colon M \to M \times M;x \mapsto (x,x)~.$$ If $M$ is a differentiable $n$-manifold the normal bundle of $\Delta_M$ is the tangent bundle $\tau_M$ of $M$. In the topological category we have: {{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 $$ (M \times M, M, \Delta_M, p_1) $$ is an $n$-dimensional microbundle, the <i>tangent microbundle</i> $\tau_M$ of $M$. {{endrem}} {{beginrem|Example}} Let $\pi \colon E \to B$ be a topological $\Rr^n$-bundle with zero section $s \colon B \to E$. Then $$(E, B, s, \pi)$$ is an $n$-dimensional microbundle. {{endrem}} {{beginthm|Definition}} 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: $$ \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \ B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \ & V_2 \ar[ur]_{j_2} } $$ {{endthm|Definition}} {{beginthm|Theorem|\cite{Kister1964|Theorem 2}}} Let $(E, B, i, j)$ be an $n$-dimensional microbundle. 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}} </wikitex> == 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.

Definition 1.1 [Milnor1964] .

An $n$ $n$ -dimensional microbundle is a quadruple $(E,B,i,j)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_s0F3jP$ $(E,B,i,j)$ such that there is a sequence

$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_EFJ92N$ $\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$

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

which makes the following diagram commute:

$\displaystyle \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_goCwRR$

For any space $M$ define the diagonal embedding

$\displaystyle \Delta_M \colon M \to M \times M;x \mapsto (x,x)~./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_QtuWpU$ $\displaystyle \Delta_M \colon M \to M \times M;x \mapsto (x,x)~.$

If $M$ is a differentiable $n$ -manifold the normal bundle of $\Delta_M$ is the tangent bundle $\tau_M$ of $M$ . In the topological category we have:

Let $M$ be topological $n$ -manifold, and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then

$\displaystyle (M \times M, M, \Delta_M, p_1)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_4bzbY8$ $\displaystyle (M \times M, M, \Delta_M, p_1)$

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

Example 1.3. Let $\pi \colon E \to B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_kC7tId$ $\pi \colon E \to B$ be a topological $\Rr^n$ $\Rr^n$ -bundle with zero section $s \colon B \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_i9nJSi$ $s \colon B \to E$ . Then

$\displaystyle (E, B, s, \pi)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_aw67yo$ $\displaystyle (E, B, s, \pi)$

is an $n$ -dimensional microbundle.

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:

$\displaystyle \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \\ B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \\ & V_2 \ar[ur]_{j_2|_{V_2}} }$

Let $(E, B, i, j)$ be an $n$ -dimensional microbundle. 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 $(E_1 \to B) \cong (E_2 \to B)$ .