Definition
The precise definition of a microbundle follows. Let B be a topological space. Then a n-microbundle consists of a triple (E, i, p) where E is a topological space (the "total space"), i is a map from B to E (the "zero section"), and p is a map from E to B ("the projection map"). Furthermore there are two conditions:
- the composition of i followed by p must be the identity;
- for every b in B, there must be a neighborhood Vb in E such that p restricted to Vb looks like a projection UB × Rn → Ub.
Note that the first condition suggests i is the zero section of a vector bundle, while the second is like the local triviality condition on a bundle. An important distinction here is that "local triviality" for microbundles only holds near a neighborhood of the zero section. E could look very wild away from that neighborhood. Also, the maps gluing together locally trivial patches of the microbundle may only overlap the fibers.
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