Microbundle

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1 Definition

The concept of a microbundle of dimension $n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}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$$B$ is a quadruple $(E,B,i,j)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_TkFPzV$$(E,B,i,j)$ where $E$$E$ is a space, $i$$i$ and $j$$j$ are maps fitting into the following diagram

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

and the following conditions hold:

1. $j\circ i=\id_B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_5Cjmd2$$j\circ i=\id_B$
2. for all $x\in B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_5gkNn6$$x\in B$ there exist open neigbourhood $U\subset B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_7GdfZa$$U\subset B$, an open neighbourhood $V\subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_9gZNqg$$V\subset E$ of $i(b)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_3rMFim$$i(b)$ and a homeomorphism
   $\displaystyle h \colon V \to U\times \mathbb{R}^n/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_33tdAs$ $$\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_7tQWiz$$(E_n,B,i_n,j_n)$, $n=1,2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_VsU9qG$$n=1,2$ over the same space $B$$B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_nX0kZN$$V_1\subset E_1$ of $i_1(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_n75fXV$$i_1(B)$ and $V_2\subset E_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_xlWel4$$V_2\subset E_2$ of $i_2(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_toq78c$$i_2(B)$ and a homeomorphism $H\colon V_1\to V_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_pXanzm$$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

The most important example of a microbundle is the micro tangent bundle 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_d4MRuw$$p_1 \colon M \times M \to M$ be the projection onto the first factor. Then

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

is an $n$$n$-dimensional microbundle, the tangent microbundle $\tau_M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1wGNDR$$\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_1SX9Q2$$\pi \colon E \to B$ be a topological $\Rr^n$$\Rr^n$-bundle with zero section $s \colon B \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_fgMJse$$s \colon B \to E$. Then

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

is an $n$$n$-dimensional microbundle.

A fundamental theorem 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_VrpLCD$$(E, B, i, j)$ be an $n$$n$-dimensional microbundle over a locally finite, finite dimensional simplicial complex $B$$B$. Then there is a neighbourhood of $i(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_d8us7Q$$i(B)$, $E_1 \subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_nXA814$$E_1 \subset E$ such that:

1. $E_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_TgXImj$$E_1$ is the total space of a topological $\Rr^n$$\Rr^n$-bundle over $B$$B$.
2. The inclusion $E_1 \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_PCph7x$$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_ZuZqiN$$E_2 \subset E$ is any other such neighbourhood of $i(B)$$i(B)$ then there is a $\Rr^n$$\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_hJkHU2$$(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