# Micro-bundle

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.