# Micro-bundle

A mapping which is a retraction (that is, there is a for which ) and which is locally trivial in the sense that for each there is a neighbourhood of in which can be represented as a direct product , with the projection onto . If for each such neighbourhood there is fixed a piecewise-linear structure in each fibre , if, moreover, the projection of on is piecewise linear and for two neighbourhoods and and any the structures on and coincide in a neighbourhood of , 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 . Namely, here , and . Each topological micro-bundle is equivalent to a unique locally trivial bundle with fibres of corresponding dimension, that is, there is a homeomorphism of some neighbourhood of in into a neighbourhood of the zero section of some bundle with fibre . 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.

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#### 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=14340