Namespaces
Variants
Actions

Difference between revisions of "Micro-bundle"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m0637501.png" /> which is a [[Retraction|retraction]] (that is, there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m0637502.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m0637503.png" />) and which is locally trivial in the sense that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m0637504.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m0637505.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m0637506.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m0637507.png" /> which can be represented as a direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m0637508.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m0637509.png" /> the projection onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375010.png" />. If for each such neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375011.png" /> there is fixed a piecewise-linear structure in each fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375012.png" />, if, moreover, the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375014.png" /> is piecewise linear and for two neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375016.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375017.png" /> the structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375019.png" /> coincide in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375020.png" />, then the micro-bundle is called piecewise linear. Other structures may be introduced similarly.
+
<!--
 +
m0637501.png
 +
$#A+1 = 32 n = 0
 +
$#C+1 = 32 : ~/encyclopedia/old_files/data/M063/M.0603750 Micro\AAhbundle
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
The notion of a micro-bundle was introduced in order to define an analogue of the [[Tangent bundle|tangent bundle]] for a topological or piecewise-linear manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375021.png" />. Namely, here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375024.png" />. Each topological micro-bundle is equivalent to a unique locally trivial bundle with fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375025.png" /> of corresponding dimension, that is, there is a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375026.png" /> of some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375029.png" /> into a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375030.png" /> of the zero section of some bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375031.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063750/m06375032.png" />. This fact is also true for piecewise-linear micro-bundles. Despite the fact that, because of this theorem, the notion of a micro-bundle has lost its theoretical interest, it is still used in concrete problems.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
 +
A mapping  $  p :  E \rightarrow X $
 +
which is a [[Retraction|retraction]] (that is, there is a  $  g :  X \rightarrow E $
 +
for which  $  pg = 1 _ {X} $)
 +
and which is locally trivial in the sense that for each  $  x \in X $
 +
there is a neighbourhood  $  U $
 +
of  $  g ( x) $
 +
in  $  E $
 +
which can be represented as a direct product  $  U = V \times \mathbf R  ^ {n} $,
 +
with  $  p \mid  _ {U} $
 +
the projection onto  $  V $.
 +
If for each such neighbourhood  $  U $
 +
there is fixed a piecewise-linear structure in each fibre  $  ( p \mid  _ {U} )  ^ {-} 1 ( x) $,
 +
if, moreover, the projection of  $  U $
 +
on  $  \mathbf R  ^ {n} $
 +
is piecewise linear and for two neighbourhoods  $  U _ {1} $
 +
and  $  U _ {2} $
 +
and any  $  x \in p ( U _ {1} ) \cap p ( U _ {2} ) $
 +
the structures on  $  ( p \mid  _ {U _ {1}  } )  ^ {-} 1 ( x) $
 +
and  $  ( p \mid  _ {U _ {2}  } )  ^ {-} 1 ( x) $
 +
coincide in a neighbourhood of  $  g ( x) $,
 +
then the micro-bundle is called piecewise linear. Other structures may be introduced similarly.
  
 +
The notion of a micro-bundle was introduced in order to define an analogue of the [[Tangent bundle|tangent bundle]] for a topological or piecewise-linear manifold  $  N $.
 +
Namely, here  $  E = N \times N $,
 +
$  p ( x , y ) = y $
 +
and  $  g ( x) = ( x , x ) $.
 +
Each topological micro-bundle is equivalent to a unique locally trivial bundle with fibres  $  \mathbf R  ^ {n} $
 +
of corresponding dimension, that is, there is a homeomorphism  $  h $
 +
of some neighbourhood  $  W $
 +
of  $  g ( X) $
 +
in  $  E $
 +
into a neighbourhood  $  \overline{W}\; $
 +
of the zero section of some bundle  $  \overline{p}\; :  E \rightarrow X $
 +
with fibre  $  \mathbf R  ^ {n} $.
 +
This fact is also true for piecewise-linear micro-bundles. Despite the fact that, because of this theorem, the notion of a micro-bundle has lost its theoretical interest, it is still used in concrete problems.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Milnor,  "Microbundles, Part I"  ''Topology'' , '''3, Suppl. 1'''  (1964)  pp. 53–80</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.C. Kirby,  L.C. Siebenmann,  "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Milnor,  "Microbundles, Part I"  ''Topology'' , '''3, Suppl. 1'''  (1964)  pp. 53–80</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.C. Kirby,  L.C. Siebenmann,  "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press  (1977)</TD></TR></table>

Revision as of 08:00, 6 June 2020


A mapping $ p : E \rightarrow X $ which is a retraction (that is, there is a $ g : X \rightarrow E $ for which $ pg = 1 _ {X} $) and which is locally trivial in the sense that for each $ x \in X $ there is a neighbourhood $ U $ of $ g ( x) $ in $ E $ which can be represented as a direct product $ U = V \times \mathbf R ^ {n} $, with $ p \mid _ {U} $ the projection onto $ V $. If for each such neighbourhood $ U $ there is fixed a piecewise-linear structure in each fibre $ ( p \mid _ {U} ) ^ {-} 1 ( x) $, if, moreover, the projection of $ U $ on $ \mathbf R ^ {n} $ is piecewise linear and for two neighbourhoods $ U _ {1} $ and $ U _ {2} $ and any $ x \in p ( U _ {1} ) \cap p ( U _ {2} ) $ the structures on $ ( p \mid _ {U _ {1} } ) ^ {-} 1 ( x) $ and $ ( p \mid _ {U _ {2} } ) ^ {-} 1 ( x) $ coincide in a neighbourhood of $ g ( x) $, then the micro-bundle is called piecewise linear. Other structures may be introduced similarly.

The notion of a micro-bundle was introduced in order to define an analogue of the tangent bundle for a topological or piecewise-linear manifold $ N $. Namely, here $ E = N \times N $, $ p ( x , y ) = y $ and $ g ( x) = ( x , x ) $. Each topological micro-bundle is equivalent to a unique locally trivial bundle with fibres $ \mathbf R ^ {n} $ of corresponding dimension, that is, there is a homeomorphism $ h $ of some neighbourhood $ W $ of $ g ( X) $ in $ E $ into a neighbourhood $ \overline{W}\; $ of the zero section of some bundle $ \overline{p}\; : E \rightarrow X $ with fibre $ \mathbf R ^ {n} $. This fact is also true for piecewise-linear micro-bundles. Despite the fact that, because of this theorem, the notion of a micro-bundle has lost its theoretical interest, it is still used in concrete problems.

Comments

References

[a1] J. Milnor, "Microbundles, Part I" Topology , 3, Suppl. 1 (1964) pp. 53–80
[a2] R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977)
How to Cite This Entry:
Micro-bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Micro-bundle&oldid=47835
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article