A real microbundle of dimension is a 4-tuple where
is a topological space (the total space of ),
is a topological space (the base space of ),
and a continuous map (projection),
another continuous map (inclusion of base space)
is a section of , i.e.
the local triviality condition holds:
for all , there are neighborhoods and and a homeomorphism such that and for all . The open subspace is called the zero section of .
A morphism of microbundles is a germ of maps from neighborhoods of the zero section of to , which commutes with projections and inclusions, with composition defined for representatives as composition of functions on smaller neighborhoods.
In particular, an isomorphism of microbundles can be represented by a homeomorphism from a neighborhood of the zero section in to a neighborhood of the zero section in commuting with projections and inclusions of the zero sections.
The main example is the tangent microbundle of a topological manifold where is the projection onto the first factor. If is a chart of the manifold around point (where and is a homeomorphism with ) then define by .
David Roberts: A couple of years ago I thought of importing topological groupoids to this concept for the following reason: The tangent microbundle , when is a manifold, is the groupoid integrating the tangent bundle of . If we have a general Lie groupoid, we can form the Lie algebroid, which is a very interesting object. If we have a topological groupoid, it seems to me that there should be a microbundle-like object that acts like the algebroid of that groupoid. This should reduce to the tangent microbundle in the case of the codiscrete groupoid = pair groupoid. Perhaps not all topological groupoids would have an associated algebroid, but those wih source and target maps that are topological submersions probably will.
Microbundles were defined by John Milnor.
Useful references are for instance