Microbundle

<h2 id="definition">Definition</h2>
A (topological) 
 
 
 
 n
 
 
 {\displaystyle n}
 
-microbundle over a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
 
 
 
 B
 
 
 {\displaystyle B}
 
 (the "base space") consists of a triple 
 
 
 
 (
 E
 ,
 i
 ,
 p
 )
 
 
 {\displaystyle (E,i,p)}
 
, where 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is a topological space (the "total space"), 
 
 
 
 i
 :
 B
 →
 E
 
 
 {\displaystyle i:B\to E}
 
 and 
 
 
 
 p
 :
 E
 →
 B
 
 
 {\displaystyle p:E\to B}
 
 are <a href="/facts/Continuous_maps/sKbl02pB">continuous maps</a> (respectively, the "zero section" and the "projection map") such that:

<ol><li>the composition 
 
 
 
 p
 ∘
 i
 
 
 {\displaystyle p\circ i}
 
 is the identity of 
 
 
 
 B
 
 
 {\displaystyle B}
 
;</li>
<li>for every 
 
 
 
 b
 ∈
 B
 
 
 {\displaystyle b\in B}
 
, there are a <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighborhood</a> 
 
 
 
 U
 ⊆
 B
 
 
 {\displaystyle U\subseteq B}
 
 of 
 
 
 
 b
 
 
 {\displaystyle b}
 
 and a neighbourhood 
 
 
 
 V
 ⊆
 E
 
 
 {\displaystyle V\subseteq E}
 
 of 
 
 
 
 i
 (
 b
 )
 
 
 {\displaystyle i(b)}
 
 such that 
 
 
 
 i
 (
 U
 )
 ⊆
 V
 
 
 {\displaystyle i(U)\subseteq V}
 
, 
 
 
 
 p
 (
 V
 )
 ⊆
 U
 
 
 {\displaystyle p(V)\subseteq U}
 
, 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphic</a> to 
 
 
 
 U
 ×
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle U\times \mathbb {R} ^{n}}
 
 and the maps 
 
 
 
 
 p
 
 ∣
 V
 
 
 :
 V
 →
 U
 
 
 {\displaystyle p_{\mid V}:V\to U}
 
 and 
 
 
 
 
 i
 
 ∣
 U
 
 
 :
 U
 →
 V
 
 
 {\displaystyle i_{\mid U}:U\to V}
 
 <a href="/facts/Commutative_diagram/bdG8xuer">commute</a> with 
 
 
 
 
 
 p
 r
 
 
 1
 
 
 :
 U
 ×
 
 
 R
 
 
 n
 
 
 →
 U
 
 
 {\displaystyle \mathrm {pr} _{1}:U\times \mathbb {R} ^{n}\to U}
 
 and 
 
 
 
 U
 →
 U
 ×
 
 
 R
 
 
 n
 
 
 ,
 x
 ↦
 (
 x
 ,
 0
 )
 
 
 {\displaystyle U\to U\times \mathbb {R} ^{n},x\mapsto (x,0)}
 
.</li></ol>
In analogy with vector bundles, the <a href="/facts/Integer/PUwwrawx">integer</a> 
 
 
 
 n
 ≥
 0
 
 
 {\displaystyle n\geq 0}
 
 is also called the rank or the fibre dimension of the microbundle. Similarly, note that the first condition suggests 
 
 
 
 i
 
 
 {\displaystyle i}
 
 should be thought of as the zero <a href="/facts/Section_(fiber_bundle)/DLnpXS5a">section</a> 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 
 
 
 
 E
 
 
 {\displaystyle E}
 
 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 <a href="/facts/Category_(mathematics)/7WzxUThf">categories</a> more general than the <a href="/facts/Category_of_manifolds/fgE931uE">smooth one</a>, such as that of <a href="/facts/Piecewise_linear_manifold/dq3fTkPT">piecewise linear manifolds</a>, by replacing topological spaces and continuous maps by suitable objects and morphisms.

<h2 id="examples">Examples</h2>
<ul><li>Any <a href="/facts/Vector_bundle/cXACAa1I">vector bundle</a> 
 
 
 
 p
 :
 E
 →
 B
 
 
 {\displaystyle p:E\to B}
 
 of rank 
 
 
 
 n
 
 
 {\displaystyle n}
 
 has an obvious underlying 
 
 
 
 n
 
 
 {\displaystyle n}
 
-microbundle, where 
 
 
 
 i
 
 
 {\displaystyle i}
 
 is the zero section.</li>
<li>Given any topological space 
 
 
 
 B
 
 
 {\displaystyle B}
 
, the <a href="/facts/Cartesian_product/BfqBWzdL">cartesian product</a> 
 
 
 
 B
 ×
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle B\times \mathbb {R} ^{n}}
 
 (together with the projection on 
 
 
 
 B
 
 
 {\displaystyle B}
 
 and the map 
 
 
 
 x
 ↦
 (
 x
 ,
 0
 )
 
 
 {\displaystyle x\mapsto (x,0)}
 
) defines an 
 
 
 
 n
 
 
 {\displaystyle n}
 
-microbundle, called the standard trivial microbundle of rank 
 
 
 
 n
 
 
 {\displaystyle n}
 
. Equivalently, it is the underlying microbundle of the trivial vector bundle of rank 
 
 
 
 n
 
 
 {\displaystyle n}
 
.</li>
<li>Given a <a href="/facts/Topological_manifold/eLKXMzzR">topological manifold</a> of dimension 
 
 
 
 n
 
 
 {\displaystyle n}
 
, the cartesian product 
 
 
 
 M
 ×
 M
 
 
 {\displaystyle M\times M}
 
 together with the projection on the first component and the <a href="/facts/Diagonal_map/F5hjgRlE">diagonal map</a> 
 
 
 
 Δ
 :
 M
 →
 M
 ×
 M
 
 
 {\displaystyle \Delta :M\to M\times M}
 
 defines an 
 
 
 
 n
 
 
 {\displaystyle n}
 
-microbundle, called the tangent microbundle of 
 
 
 
 M
 
 
 {\displaystyle M}
 
.</li>
<li>Given an 
 
 
 
 n
 
 
 {\displaystyle n}
 
-microbundle 
 
 
 
 (
 E
 ,
 i
 ,
 p
 )
 
 
 {\displaystyle (E,i,p)}
 
 over 
 
 
 
 B
 
 
 {\displaystyle B}
 
 and a continuous map 
 
 
 
 f
 :
 A
 →
 B
 
 
 {\displaystyle f:A\to B}
 
, the space 
 
 
 
 
 f
 
 ∗
 
 
 E
 :=
 {
 (
 a
 ,
 e
 )
 ∈
 A
 ×
 E
 ∣
 f
 (
 a
 )
 =
 p
 (
 e
 )
 }
 
 
 {\displaystyle f^{*}E:=\{(a,e)\in A\times E\mid f(a)=p(e)\}}
 
 defines an 
 
 
 
 n
 
 
 {\displaystyle n}
 
-microbundle over 
 
 
 
 A
 
 
 {\displaystyle A}
 
, called the pullback (or induced) microbundle by 
 
 
 
 f
 
 
 {\displaystyle f}
 
, together with the projection 
 
 
 
 p
 :=
 
 
 p
 r
 
 
 1
 
 
 :
 
 f
 
 ∗
 
 
 E
 →
 A
 
 
 {\displaystyle p:=\mathrm {pr} _{1}:f^{*}E\to A}
 
 and the zero section 
 
 
 
 i
 :
 A
 →
 
 f
 
 ∗
 
 
 E
 ,
 x
 ↦
 (
 x
 ,
 (
 i
 ∘
 f
 )
 (
 x
 )
 )
 
 
 {\displaystyle i:A\to f^{*}E,x\mapsto (x,(i\circ f)(x))}
 
. If 
 
 
 
 p
 :
 E
 →
 B
 
 
 {\displaystyle p:E\to B}
 
 is a vector bundle, the pullback microbundle of its underlying microbundle is precisely the underlying microbundle of the standard <a href="/facts/Pullback_bundle/nBdt2Cx7">pullback bundle</a>.</li>
<li>Given an 
 
 
 
 n
 
 
 {\displaystyle n}
 
-microbundle 
 
 
 
 (
 E
 ,
 i
 ,
 p
 )
 
 
 {\displaystyle (E,i,p)}
 
 over 
 
 
 
 B
 
 
 {\displaystyle B}
 
 and a subspace 
 
 
 
 A
 ⊆
 B
 
 
 {\displaystyle A\subseteq B}
 
, the restricted microbundle, also denoted by 
 
 
 
 
 E
 
 ∣
 A
 
 
 =
 
 p
 
 −
 1
 
 
 (
 A
 )
 
 
 {\displaystyle E_{\mid A}=p^{-1}(A)}
 
, is the pullback microbundle with respect to the inclusion 
 
 
 
 A
 ↪
 B
 
 
 {\displaystyle A\hookrightarrow B}
 
.</li></ul>
<h2 id="morphisms">Morphisms</h2>
Two 
 
 
 
 n
 
 
 {\displaystyle n}
 
-microbundles 
 
 
 
 (
 
 E
 
 1
 
 
 ,
 
 i
 
 1
 
 
 ,
 
 p
 
 1
 
 
 )
 
 
 {\displaystyle (E_{1},i_{1},p_{1})}
 
 and 
 
 
 
 (
 
 E
 
 2
 
 
 ,
 
 i
 
 2
 
 
 ,
 
 p
 
 2
 
 
 )
 
 
 {\displaystyle (E_{2},i_{2},p_{2})}
 
 over the same space 
 
 
 
 B
 
 
 {\displaystyle B}
 
 are isomorphic (or equivalent) if there exist a neighborhood 
 
 
 
 
 V
 
 1
 
 
 ⊆
 
 E
 
 1
 
 
 
 
 {\displaystyle V_{1}\subseteq E_{1}}
 
 of 
 
 
 
 
 i
 
 1
 
 
 (
 B
 )
 
 
 {\displaystyle i_{1}(B)}
 
 and a neighborhood 
 
 
 
 
 V
 
 2
 
 
 ⊆
 
 E
 
 2
 
 
 
 
 {\displaystyle V_{2}\subseteq E_{2}}
 
 of 
 
 
 
 
 i
 
 2
 
 
 (
 B
 )
 
 
 {\displaystyle i_{2}(B)}
 
, together with a <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a> 
 
 
 
 
 V
 
 1
 
 
 ≅
 
 V
 
 2
 
 
 
 
 {\displaystyle V_{1}\cong V_{2}}
 
 commuting with the projections and the zero sections.
More generally, a morphism between microbundles consists of a <a href="/facts/Germ_(mathematics)/g28mtBu5">germ</a> of continuous maps 
 
 
 
 
 V
 
 1
 
 
 →
 
 V
 
 2
 
 
 
 
 {\displaystyle V_{1}\to V_{2}}
 
 between neighbourhoods of the zero sections as above. 
An 
 
 
 
 n
 
 
 {\displaystyle n}
 
-microbundle is called trivial if it is isomorphic to the standard trivial microbundle of rank 
 
 
 
 n
 
 
 {\displaystyle n}
 
. The local triviality condition in the definition of microbundle can therefore be restated as follows: for every 
 
 
 
 b
 ∈
 B
 
 
 {\displaystyle b\in B}
 
 there is a neighbourhood 
 
 
 
 U
 ⊆
 B
 
 
 {\displaystyle U\subseteq B}
 
 such that the restriction 
 
 
 
 
 E
 
 ∣
 U
 
 
 
 
 {\displaystyle E_{\mid U}}
 
 is trivial.
Analogously to <a href="/facts/Parallelizable_manifold/CQq7nQsJ">parallelisable smooth manifolds</a>, a topological manifold is called topologically parallelisable if its tangent microbundle is trivial.

<h2 id="properties">Properties</h2>
A theorem of James Kister and <a href="/facts/Barry_Mazur/4tGbAnCX">Barry Mazur</a> states that there is a neighborhood of the zero section which is actually a <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundle</a> with fiber 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 and <a href="/facts/Structure_group/SJ7Ek6cI">structure group</a> 
 
 
 
 Homeo
 ⁡
 (
 
 
 R
 
 
 n
 
 
 ,
 0
 )
 
 
 {\displaystyle \operatorname {Homeo} (\mathbb {R} ^{n},0)}
 
, the group of homeomorphisms of 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 fixing the origin. This neighborhood is unique up to <a href="/facts/Homotopy/1nWrPxdj">isotopy</a>. Thus every microbundle can be refined to an actual fiber bundle in an essentially unique way.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
Taking the fiber bundle contained in the tangent microbundle 
 
 
 
 (
 M
 ×
 M
 ,
 Δ
 ,
 
 p
 r
 
 )
 
 
 {\displaystyle (M\times M,\Delta ,\mathrm {pr} )}
 
 gives the topological tangent bundle. Intuitively, this bundle is obtained by taking a system of small charts for 
 
 
 
 M
 
 
 {\displaystyle M}
 
, letting each chart 
 
 
 
 U
 
 
 {\displaystyle U}
 
 have a fiber 
 
 
 
 U
 
 
 {\displaystyle U}
 
 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 <a href="/facts/Robion_Kirby/QMxyDhtv">Robion Kirby</a> and <a href="/facts/Laurent_C._Siebenmann/8SKMhs5J">Laurent C. Siebenmann</a> on <a href="/facts/Smooth_structure/lfvfhqxr">smooth structures</a> and <a href="/facts/PL_structure/dq3fTkPT">PL structures</a> on <a href="/facts/Dimension/0ktmUyac">higher dimensional</a> manifolds.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<ul><li>Gauld, David; Greenwood, Sina (2000). <a href="https://doi.org/10.1090%2Fs0002-9939-00-05343-0">"Microbundles, manifolds and metrisability"</a>. <a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a>. 128 (9): 2801–2808. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fs0002-9939-00-05343-0">10.1090/s0002-9939-00-05343-0</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1664358">1664358</a>.</li>
<li>Switzer, Robert M. (2002). Algebraic topology—homotopy and homology. Classics in Mathematics. Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-42750-6. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1886843">1886843</a>. See Chapter 14.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.map.mpim-bonn.mpg.de/Microbundle">Microbundle</a> at the Manifold Atlas.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Milnor, John Willard (1964). "Microbundles. I". Topology. 3: 53–80. doi:10.1016/0040-9383(64)90005-9. MR 0161346. <a href="/wiki/John_Milnor" target="_blank">/wiki/John_Milnor</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Kister, James M. (1964). "Microbundles are fibre bundles". Annals of Mathematics. 80 (1): 190–199. doi:10.2307/1970498. JSTOR 1970498. MR 0180986. <a href="http://projecteuclid.org/euclid.bams/1183525710" target="_blank">http://projecteuclid.org/euclid.bams/1183525710</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kirby, Robion C.; Siebenmann, Laurent C. (1977). Foundational essays on topological manifolds, smoothings, and triangulations (PDF). Annals of Mathematics Studies. Vol. 88. Princeton, N.J.: Princeton University Press. ISBN 0-691-08191-3. MR 0645390. <a href="0-691-08191-3" target="_blank">0-691-08191-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Microbundle open-in-new

Microbundle