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.

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)