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{{Short description|Generalization of the concept of vector bundle}}
In [[mathematics]], a '''microbundle''' is a generalization of the concept of [[vector bundle]], introduced by the [[United States|American]] [[mathematician]] [[John Milnor]] in 1964.<ref>{{cite journal | last1=Milnor | first1=John Willard | author1-link=John Milnor | title=Microbundles. I | doi=10.1016/0040-9383(64)90005-9 |mr=0161346 | year=1964 | journal=[[Topology (journal)|Topology]] | volume=3 | pages=53–80| doi-access= }}</ref> It allows the creation of bundle-like objects in situations where they would not ordinarily be thought to exist. For example, the [[tangent bundle]] is defined for a [[smooth manifold]] but not a [[topological manifold]]; use of microbundles allows the definition of a ''topological'' tangent bundle.

==Definition==

A (topological) '''''<math>n</math>''-microbundle''' over a [[topological space]] ''<math>B</math>'' (the "base space") consists of a triple <math>(E, i, p)</math>, where ''<math>E</math>'' is a topological space (the "total space"), ''<math>i: B \to E</math>'' and ''<math>p: E \to B</math>'' are [[continuous maps]] (respectively, the "zero section" and the "projection map") such that:

#the composition ''<math>p \circ i</math>'' is the identity of ''<math>B</math>'';
#for every ''<math>b \in B</math>'', there are a [[Neighbourhood (mathematics)|neighborhood]] <math>U \subseteq B</math> of <math>b</math> and a neighbourhood <math>V \subseteq E</math> of <math>i(b)</math> such that <math>i(U) \subseteq V</math>, <math>p(V) \subseteq U</math>, <math>V</math> is [[Homeomorphism|homeomorphic]] to <math>U\times \R^n</math> and the maps ''<math>p_{\mid V}: V \to U</math>'' and ''<math>i_{\mid U}: U \to V</math>'' [[Commutative diagram|commute]] with ''<math>\mathrm{pr}_1: U \times \mathbb{R}^n \to U</math>'' and ''<math>U \to U \times \mathbb{R}^n, x \mapsto (x,0)</math>''.
In analogy with vector bundles, the [[integer]] '''''<math>n \geq 0</math>''''' is also called the '''rank''' or the fibre dimension of the microbundle. Similarly, note that the first condition suggests ''<math>i</math>'' should be thought of as the zero [[Section (fiber bundle)|section]] of a vector bundle, while the second mimics the local triviality condition on a bundle. An important distinction here is that "local triviality" for microbundles only holds near a neighborhood of the zero section. The space ''<math>E</math>'' could look very wild away from that neighborhood. Also, the maps gluing together locally trivial patches of the microbundle may only overlap the fibers.

The definition of microbundle can be adapted to other [[Category (mathematics)|categories]] more general than the [[Category of manifolds|smooth one]], such as that of [[Piecewise linear manifold|piecewise linear manifolds]], by replacing topological spaces and continuous maps by suitable objects and morphisms.

== Examples ==

* Any [[vector bundle]] <math>p: E \to B</math> of rank <math>n</math> has an obvious '''underlying <math>n</math>-microbundle''', where <math>i</math> is the zero section.
* Given any topological space ''<math>B</math>'', the [[cartesian product]] <math>B \times \mathbb{R}^n</math> (together with the projection on ''<math>B</math>'' and the map ''<math>x \mapsto (x,0)</math>'') defines an ''<math>n</math>''-microbundle, called the '''standard trivial microbundle''' of rank <math>n</math>. Equivalently, it is the underlying microbundle of the trivial vector bundle of rank <math>n</math>.
* Given a [[topological manifold]] of dimension <math>n</math>, the cartesian product <math>M\times M</math> together with the projection on the first component and the [[diagonal map]] <math>\Delta: M\to M\times M</math> defines an ''<math>n</math>''-microbundle, called the '''tangent microbundle''' of <math>M</math>.
* Given an ''<math>n</math>''-microbundle <math>(E, i, p)</math> over ''<math>B</math>'' and a continuous map <math>f: A \to B</math>, the space <math>f^*E := \{ (a,e) \in A \times E \mid f(a)=p(e) \}</math> defines an ''<math>n</math>''-microbundle over <math>A</math>, called the '''pullback (or induced) microbundle''' by <math>f</math>, together with the projection ''<math>p:= \mathrm{pr}_1: f^*E \to A</math>'' and the zero section ''<math>i: A \to f^*E, x \mapsto (x, (i \circ f)(x))</math>''. If <math>p: E \to B</math> is a vector bundle, the pullback microbundle of its underlying microbundle is precisely the underlying microbundle of the standard [[pullback bundle]].
* Given an ''<math>n</math>''-microbundle <math>(E, i, p)</math> over ''<math>B</math>'' and a subspace <math>A \subseteq B</math>, the '''restricted microbundle''', also denoted by <math>E_{\mid A} = p^{-1}(A)</math>, is the pullback microbundle with respect to the inclusion <math>A \hookrightarrow B</math>.

== Morphisms ==
Two <math>n</math>-microbundles <math>(E_1, i_1, p_1)</math> and <math>(E_2, i_2, p_2)</math> over the same space <math>B</math> are '''isomorphic''' (or equivalent) if there exist a neighborhood <math>V_1 \subseteq E_1</math> of <math>i_1(B)</math> and a neighborhood <math>V_2 \subseteq E_2</math> of <math>i_2(B)</math>, together with a [[homeomorphism]] <math>V_1 \cong V_2</math> commuting with the projections and the zero sections.

More generally, a '''morphism''' between microbundles consists of a [[Germ (mathematics)|germ]] of continuous maps <math>V_1 \to V_2</math> between neighbourhoods of the zero sections as above.

An <math>n</math>-microbundle is called '''trivial''' if it is isomorphic to the standard trivial microbundle of rank <math>n</math>. The local triviality condition in the definition of microbundle can therefore be restated as follows: for every ''<math>b \in B</math>'' there is a neighbourhood ''<math>U \subseteq B</math>'' such that the restriction <math>E_{\mid U}</math> is trivial.

Analogously to [[Parallelizable manifold|parallelisable smooth manifolds]], a topological manifold is called '''topologically parallelisable''' if its tangent microbundle is trivial.

==Properties==

A theorem of James Kister and [[Barry Mazur]] states that there is a neighborhood of the zero section which is actually a [[fiber bundle]] with fiber <math>\R^n</math> and [[structure group]] <math>\operatorname{Homeo}(\R^n,0)</math>, the group of homeomorphisms of <math>\R^n</math> fixing the origin. This neighborhood is unique up to [[Homotopy#Isotopy|isotopy]]. Thus every microbundle can be refined to an actual fiber bundle in an essentially unique way.<ref>{{cite journal | last=Kister | first=James M. | title=Microbundles are fibre bundles | journal=[[Annals of Mathematics]] | volume=80 | issue=1 | year=1964 | doi=10.2307/1970498 | pages=190–199 | jstor=1970498 | mr=0180986| url=http://projecteuclid.org/euclid.bams/1183525710 }}</ref>

Taking the fiber bundle contained in the tangent microbundle <math>(M\times M, \Delta, \mathrm{pr})</math> gives the '''topological tangent bundle'''. Intuitively, this bundle is obtained by taking a system of small charts for ''<math>M</math>'', letting each chart ''<math>U</math>'' have a fiber ''<math>U</math>'' over each point in the chart, and gluing these trivial bundles together by overlapping the fibers according to the transition maps.

Microbundle theory is an integral part of the work of [[Robion Kirby]] and [[Laurent C. Siebenmann]] on [[smooth structure]]s and [[PL structure]]s on [[Dimension#Manifolds|higher dimensional]] manifolds.<ref>{{cite book|first1= Robion C. |last1=Kirby |author1-link=Robion Kirby|first2= Laurent C.|last2= Siebenmann| author2-link=Laurent C. Siebenmann|title=Foundational essays on topological manifolds, smoothings, and triangulations| url=http://www.maths.ed.ac.uk/~aar/papers/ks.pdf| series=Annals of Mathematics Studies|volume= 88| publisher=[[Princeton University Press]]| location=Princeton, N.J.|year= 1977|isbn= 0-691-08191-3|mr=0645390}}</ref>

==References==
<references/>
* {{cite journal | last1=Gauld | first1=David | last2=Greenwood | first2=Sina | title=Microbundles, manifolds and metrisability| journal=[[Proceedings of the American Mathematical Society]] | volume=128 | issue=9 | year=2000 | doi=10.1090/s0002-9939-00-05343-0 | pages=2801–2808 | mr=1664358| doi-access=free }}
* {{cite book | last1=Switzer | first1=Robert M. | title=Algebraic topology—homotopy and homology | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-42750-6 |mr=1886843 | year=2002}} See Chapter 14.

== External links ==
*[http://www.map.mpim-bonn.mpg.de/Microbundle Microbundle] at the Manifold Atlas.

[[Category:Geometric topology]]
[[Category:Algebraic topology]]

Microbundle - Revision history

imported>OAbot: Open access bot: doi updated in citation with #oabot.