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.
Useful references are for instance