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.

This page has not been refereed. The information given here might be incomplete or provisional.

The user responsible for this page is Matthias Kreck. No other user may edit this page at present.

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 Kister [Kister1964], and independently Mazur, showed that every microbundle uniquely determines a topological $\Rr^n/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_xUJjuu$$\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/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_b8uLOm$$B$ be a topological space. An $n$$n$-dimensional microbundle over $B$$B$ is a quadruple $(E,B,i,j)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_9aSYyf$$(E,B,i,j)$ where $E$$E$ is a space, $i$$i$ and $j$$j$ are maps fitting into the following diagram

$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_hyxbJ8$ $$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$$

and the following conditions hold:

1. $j\circ i=\id_B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_pRHYj2$$j\circ i=\id_B$.
2. For all $x\in B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1ALOnW$$x\in B$ there exist open neigbourhood $U\subset B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_3mdrUQ$$U\subset B$, an open neighbourhood $V\subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_PCceQL$$V\subset E$ of $i(b)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_9ETQdH$$i(b)$ and a homeomorphism
   $\displaystyle h \colon V \to U\times \mathbb{R}^n/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_LnbS0C$ $$\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_7KEkez$$(E_n,B,i_n,j_n)$, $n=1,2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_ZBC4Rv$$n=1,2$ over the same space $B$$B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_dZeAWs$$V_1\subset E_1$ of $i_1(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_xEU0qq$$i_1(B)$ and $V_2\subset E_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_hUZvlo$$V_2\subset E_2$ of $i_2(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_VJpwFm$$i_2(B)$ and a homeomorphism $H\colon V_1\to V_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_JpTipl$$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_vtalHk$$p_1 \colon M \times M \to M$ be the projection onto the first factor. Then

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

is an $n$$n$-dimensional microbundle, the tangent microbundle $\tau_M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_7N4fzk$$\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_nwD09k$$\pi \colon E \to B$ be a topological $\Rr^n$$\Rr^n$-bundle with zero section $s \colon B \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_3HKfam$$s \colon B \to E$. Then the quadruple

$\displaystyle (E, B, s, \pi)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_5BoCzn$ $$\displaystyle (E, B, s, \pi)$$

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

A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem, proven independently by Kister and Mazur.

Theorem 2.4 [Kister1964, Theorem 2]. Let $(E, B, i, j)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_frO7np$$(E, B, i, j)$ be an $n$$n$-dimensional microbundle over a locally finite, finite dimensional simplicial complex $B$$B$. Then there is a neighbourhood of $i(B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_h1lmCr$$i(B)$, $E_1 \subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_PAypfu$$E_1 \subset E$ such that:

1. $E_1/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vxsBhx$$E_1$ is the total space of a topological $\Rr^n$$\Rr^n$-bundle over $B$$B$.
2. The inclusion $E_1 \to E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_NNntJA$$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_REkwAE$$E_2 \subset E$ is any other such neighbourhood of $i(B)$$i(B)$ then there is a $\Rr^n$$\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Fa1xQI$$(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.