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.
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) |
Micro-bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Micro-bundle&oldid=14340