View source for Micro-bundle

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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====

====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>